NORTH. HOLLAND
MATHEMATICS STUDIES
Spectra I Theory of Differential Operators
I.W. KNOWLES R.T.LEWlS Editors
NORTH·HOlLAND
55
SPECTRAL THEORY OF DIFFERENTIAL OPERATORS
This page intentionally left blank
This Page Intentionally Left Blank
© North-Holland
Publishing Company,1981
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, ortransmitted, in any form or by.any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner.
ISBN: 0444 86277 3
Publishers: NORTH-HOLLAND PUBLISHING COMPANY AMSTERDAM NEW YORK . OXFORD Sole distributorsfor the U.S.A. and Canada:
ELSEVIER NORTH-HOLLAND, INC. 52 VANDERBILT AVENUE, NEW YORK, N.Y. 10017
PRINTED IN THE NETHERLANDS
This volume is respectfully dedicated to Professor F.V. Atkinson on the ocassion of his sixty-fifth birthday.
This page intentionally left blank
This Page Intentionally Left Blank
NORTH-HOLLAND MATHEMATICS STUDIES
Spectral Theory of Differential Operators Proceedings ofthe Conference held at the University of Alabama in Birmingham, Birmingham,Alabama, U.S.A., March 26-28, 1981
Edited by
IAN W. KNOWLES and
ROGER T. LEWIS University of Alabama Birmingham, Alabama, U.S.A.
19]1
N.H 1981
q~c
NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM. NEW YORK. OXFORD
55
PREFACE This volume forms a permanent record of lectures given at the International Conference on Spectral Theory of Differential Operators held at the University of Alabama in Birmingham March 26-28, 1981. The conference was supported by about 90 mathematicians from North America and Europe. Its main purpose was to provide a forum for the discussion of recent work in certain areas of the theory of ordinary and partial differential equations loosely connected under the general heading of Spectral Theory. Invited one-hour plenary lectures were given by F. V. Atkinson, who gave a series of three lectures, P. Deift, \~. N. Everitt, H. Ka If, T. Kato, R. M. Kauffman, M. Schechter and B. Simon. The remainder of the programme consisted of invited special session lectures, each of one-half hour duration. On behalf of the participants, the conference directors acknowledge, with gratitude, the generous financial support provided by the School of Natural Sciences and Mathematics and the School of Graduate Studies of the University of Alabama in Birmingham. \~e are especially grateful to Professor Peter V. O'Neil, Chairman of the Department of Mathematics, for his support and encouragement. Without this support the conference could not have taken place. We acknowledge also the valuable support provided by the faculty and staff of the Department of Mathematics. Here, we are particularly grateful to Professor Fred Martens, for his efficient direction of the local arrangements, and to Mrs. Eileen Schauer for her speedy and expert typing of much of the conference material, including many of the articles appearing in this volume. Finally, it is a pleasure to acknowledge the friendly assistance of Drs. Arjen Sevenster, Editor of the Mathematics Studies Series of North-Holland, during the preparation of these Proceedings. Ian \1. Knowles Roger T. Lewi s Conference Directors
vii
This page intentionally left blank
This Page Intentionally Left Blank
CONTENTS C. D. Ahlbrandt, D. B. Hinton and R. T. Lewis Transformations of ordinary differential operators W. All egretto
Finiteness criteria for the negative spectrum and nonoscillation theory for a class of higher order elliptic operators
9
F. V. Atkinson
A class of limit-point criteria
13
M. F. Ba rns 1ey
Bounds for the linearly perturbed eigenvalue problem
37
M. F. Barnsley, J. V. Herod, D. L. Mosher and G. B. Passty Analysis of Boltzmann equations in Hilbert space by means of a non-linear eigenvalue property
45
John Baxl ey Christer Bennewitz Richard C. Brown
Robert Ca rro 11 J. M. Combes and R. Weder
Constantin Corduneanu A. Devinatz and P. Rejto W. N. Everitt M. Faierman
J. Fleckinger
Some partial differential operators with discrete spectra
53
Spectral theory for hermitean differential systems
61
Wirtinger inequalities, Dirichlet functional inequalities, and the spectral theory of linear operators and relations
69
A survey of some recent results in transmutation
81
Spectral theory and unbounded obstacle scattering
93
Almost periodic solutions for infinite delay systems
99
A Schrodinger operator with an oscillating potential
107
On certain regular ordinary differential expressions and related operators
115
An eigenfunction expansion associated with a two-parameter system of differential equations
169
Distribution of eigenvalues of operators of Schrodinger type
173
ix
x
CONTENTS
The local asymptotics of continuum eigenfunction expansions
181
Some open problems on asymptotics of m-coefficients
189
Singular linear ordinary differential equations with non-zero second auxiliary polynomial
193
R. Kent Goodrich and Karl Gustafson Higher dimensional spectral factorization with applications to digital filtering
199
J. R. Graef and P. W. Spikes The limit point-limit circle problem for nonlinear equations
207
Stephen Fulling Charles T. Fulton Richard C. Gilbert
1som H. Herron
A model problem for the linear stability of nearly parallel flows
Don B. Hinton and K. Shaw Titchmarsh-Weyl theory for Hamiltonian systems
211
219
Christopher Hunter
Two parametric eigenvalue problems of differential equations
233
Arne Jensen
Schrodinger operators in the low energy 1imit: some recent results in L2(R4)
243
Hans G. Kaper
Long-time behaviour of a nuclear reactor
247
Tosio Kato
Remarks on the selfadjointness and related problems for differential operators
253
R. M. Kauffman
A Weyl theory for a class of elliptic boundary value problems on a half-space
267
Ian W. Knowles and O. Race
On the correctness of boundary conditions for certain linear differential operators
279
S. J. Lee
Index and nonhomogeneous conditions for linear manifolds
289
Howard A. Levine
On the positive spectrum of Schrodinger operators with long range potentials
295
Roger T. Lewi s
The spectra of some singular elliptic operators of second order
303
Peter McCoy
Recapturing solutions of an elliptic partial differential equation
319
Joyce McLaughlin
Fourth order inverse eigenvalue problems
327
Angelo B. Mingarelli
Sturm theory in n-space
337
Branko Najman
Selfadjointness of matrix operators
343
A. G. Ramm
Spectral properties of some nonselfadjoint operators and some applications
349
CONTENTS
xi
Thomas T. Read
Dirichlet solutions of fourth order differential equations
355
Martin Schechter
Spectral and scattering theory for propagative systems
361
B. Simon
Spectral analysis of multiparticle Schrodinger operators. Schrodinger operators with almost periodic potentials
369
Estimates for eigenvalues of the Laplacian on compact Riemannian manifolds
371
Phil ip Wal ker
The square-integrable span of locally square integrable functions
375
Stephen D. Wray
On a conditionally convergent Dirichlet integral associated with a differential expression
379
Udo Simon
LECTURES NOT APPEARING IN PROCEEDINGS H. E. Benzinger
Rayleigh-Schrodinger perturbation of semi-groups
C. Bill igheimer
Spectral propertiei of differential operators in the complex plane in B -algebras
P. J. Browne
Eigencurve asymptotics for two parameter eigenvalue problems
H. L. Cycon
On the form sum and the Friedrichs extension of Schrodinger operators with singular potentials
P. Deift
New results in inverse theory
E.
Harrell
H. Kalf
Very small spectral properties of Schrodinger operators
J. Neuberger
On the non-existence of eigenvalues of Dirac operators Operators on L2 (I) ~ Cm Calculation of eigenvalues for -~ + V on a region in R3
S. Ranki n
Generation and representation of cosine families
B. Textorius
Generalized resolvents and resolvent matrices of canonical differential relations in Hilbert space
R. R. D. Kemp
xii
ADDRESS LIST OF CONTRIBUTORS C. D. Ahlbrandt W. All egretto F. V. Atkinson M. F. Barnsley
John Baxl ey Christer Bennewitz C. Bi11igheimer Richard C. Brown Robert Carroll Constantin Corduneanu Percy Deift Allen Devinatz W. N. Everitt M.
Faierman
J. Fleckinger Stephen Fu 11 i ng Charles T. Fulton Richard C. Gilbert R. Kent Goodrich
Department of Mathematics, University of Missouri, Columbia, Missouri 65211 Department of Mathematics, University of Alberta, Edmonton, CANADA T6G 2Gl Department of Mathematics, University of Toronto, Toronto, CANADA M5S lAl School of Mathematics, Georgia Institute of Technolog~ Atlanta, Georgia 30332 Department of Mathematics, Wake Forest University, Winston Salem, North Carolina 27109 Department of Mathematics, University of Uppsala, Uppsala, SWEDEN Department of ~·1athematics, McMaster University, Hamilton, Ontario, CANADA L8S 4Kl Department of Mathematics, University of Alabama (Tuscaloosa), University, Alabama 35486 Department of Mathematics, University of Illinois, Urbana, Illinois 61801 Department of Mathematics, University of Texas at Arlington, Arlington, Texas 76019 Courant Institute, New York University, 251 Mercer St., New York, New York 10012 Department of Mathematics, Northwestern University, Evanston, Illinois 60091 Department of Mathematics, The University of Dundee, Dundee, SCOTLAND, UNITED KINGDOM DDl 4HN Department of Mathematics, University of the Witwatersrand, Johannesburg, 2001 SOUTH AFRICA Universite Paul Sabatier, 118, Route de Narbonne, 118 31062 Toulouse CEDEX FRANCE Department of Mathematics, Texas A & M University, College Station, Texas 77843 Mathematics Department, Penn State University, University Park, Pennsylvania 16802 Department of Mathematics, California State University, Fullerton, Fullerton, California 92634 Department of r~athematics, University of Colorado, Boulder, Colorado 80309 xiii
xiv
Karl Gustafson James V. Herod Isom H. Herron Don B. Hinton Christopher Hunter Arne Jensen Hans G. Kaper Tosio Kato R. M. Kauffman Ian W. Knowles Luis Kramarz S. J. Lee Howard A. Levine Roger T. Lewi s Peter McCoy Joyce McLaughlin Angelo Mingarelli David Mosher Branko Najman Gregory B. Passty A. G. Ramm Thomas T. Read Martin Schechter Ken Shaw B. Simon
LIST OF CONTRIB UTORS
Department of Mathematics, University of Colorado, Boulder, Colorado 80309 School of Mathematics, Georgia Institute of Technolog~ Atlanta, Georgia 30332 Department of Mathematics, Howard University, Washington, D. C. 20059 Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37916 Department of Mathematics, Florida State University, Tallahassee, Florida 32306 Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506 Argonne National Laboratory, Argonne, Illinois 60439 Department of Mathematics, University of California at Berkeley, Berkeley, California 94720 Department of Mathematics, Western Washington University, Bellingham, Washington 98225 Department of Mathematics, University of Alabama in Birmingham, Birmingham, Alabama 35294 r~athematics Department, Emory University, Atlanta, Georgia 30322 Department of Mathematics, Pan American University, Edinburg, Texas 78539 Department of Mathematics, Iowa State University Ames, Iowa 50010 Department of Mathematics, University of Alabama in Birmingham, Birmingham, Alabama 35294 United States Naval Academy, Annapolis, Maryland 21402 Department of Mathematics, Rensselaer Polytechnic Institute, Troy, New York 12181 Department of Mathematics, University of Ottawa, Ottawa, Ontario, CANADA K1N 9B4 School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332 Department of Mathematics, University of California, Berkeley, California 94720 School of Mathematics, Georgia Institute of TechnolQgY, Atlanta, Georgia 30332 -Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109 Department of Mathematics, Western Washington University, Bellingham, Washington 98225 Division of Natural Sciences and Mathematics, Yeshiva University, 2495 Amsterdam Avenue, New York, NY 10033 Department of Mathematics, V. P. I., Blacksburg, Vi rgi ni a 24061 Department of Mathematics, California Institute of Technology, Pasadena, California 91125
LIST OF CONTRIB UTORS
Udo Simon Paul W. Spikes Philip Walker Ri ca rdo vJeder Stephen D. Wray
xv
Technische Universitat Berlin, l-Berlin 12, FRG, WEST GERMANY Department of Mathematics, Mississippi State University, Mississippi State, Mississippi 39762 Department of Mathematics, University of Houston, Houston, Texas 77004 Instituto de Investigaciones en Matematicas Aplicadas y en Sistemas, Universidag Nacional Autonoma de Mexico, Apartado Postal 20-726, MEXICO 20, D. F. Department of Mathematics and Computer Science, Mount Allison University, Sackville, New Brunswick, CANADA EOA 3CO
This page intentionally left blank
This Page Intentionally Left Blank
Spectral Theory of Differential Operators I. IN. Knowles and R. T. Lewis leds.) © North·Holland Publishing Company, 1981
TRANSFORMATIONS OF ORDINARY DIFFERENTIAL OPERATORS Calvin D. Ahlbrandt
Don B. Hinton
Univ. of Missouri Columbia, MO 65211 U.S.A.
Univ. of Tennessee Knoxville, TN 37916 U.S.A.
Kummer-Liouville coordinate changes order vector differential operators form. This study is preliminary to forms and transformation theory for differential operators.
1.
Roger T. Lewis* Univ. of Ala. in Birmingham Birmingham, AL 35294 U.S.A.
are presented for fourth of the formally self-adjoint the development of canonical linear fourth order partial
INTRODUCTION
This is part of an ongoing investigation of variable change methods for differential operators. The impetus for the general study was a desire to unify results in spectral and oscillation theories for operators having a singularity at 0 and operators having a singularity at The transformation theory for scalar ordinary operators of even order was developed for the real case in [2]. More general results for the second order, including certain partial differential operators, were presented in [3]. An extension of the "Kelvin transformation" to powers of the Laplacian was presented in [4] and a discussion of various equivalences of operators was given in [1]. The present fourth order vector discussion illustrates the theory for higher order vector ordinary differential operators and builds notation for the fourth order partial case. The transformation theory for the odd order cases is obtained as a corollary to the even order cases. 2.
THE SECOND ORDER CASE
consider the second order scalar Jacobi-Reid [8] canonical form L[y]
=
-(r(x)y' + q(x)y)' + (q(x)y' + p(x)y).
(2.1)
Suppose that p and r are real valued and q is complex valued on a real interval x. The special case where q is real valued arises in the Calculus of Variations [6]. I f p, q, and r are continuous and r never vanishes on x, then the "off-diagonal" terms may be removed by a variable change to produce a two term operator [3,7]
=
L[y]
(2.2)
-(r(x)y')' + p(x)y.
However, the form given in (2.1) has several advantages over the form (2.2). First, the form of (2.1) is preserved under Kummer-Liouville coordinate changes [3, TH. 2.2] y(x)
=
~(x)z(t),
t
=
f(x),
~
and
f'
nonvanishing,
(2.3)
with ~ complex valued. (The form of (2.2) is preserved in case ~ is real valued, but not necessarily if ~ is complex valued.) Second, the generalization of (2.1) to the vector case [8] L[y]
=
-(R(x)y' + Q(x)y)' + (Q*(x)y' + P(x)y,
(2.4)
(here P, Q, and Rare n x n complex matrix valued with P and R hermitian on X), includes a useful first order case. Indeed, the special case of (2.4)
2
G.D. AHLBRANDT et al.
with P hermitian, R - 0, and the "Atkinson form" [5]
Q
L[y]
a constant skew hermitian matrix reduces to
= Jy '
+ P(x)y
(2.5)
for J defined as 2Q*. Third, the form of (2.4) is needed for general KummerLiouville transformations
= H(x)z(t),
y(x)
=
t
f(x),
H nonsingular,
f'
~
of (2.4) even if all the involved matrices have real entries. of (2.4) under (2.6) is of the form [3] LO[Z]
0,
(2.6) The image operator
-(ROZ' + QOz)' + (QOz' + Poz)
=
with the coefficient matrices being functions of t conditions on P, Q, R, H, and f the operators the identi ty
on L
(2.7)
T = f (X). Under certain and LO ·are related by
{(l/!f'!)H*L[y]}(x) = LO[Z](t) for
(x,y)
and
(t,z)
(2.8)
related by (2.6).
A generalization of the "Atkinson form" to partial differential operators was included in [3]. 3.
A CANONICAL FORM FOR FOURTH ORDER ORDINARY OPERATORS
If the vector operator L[y]
(R (x)y")
",
R* (x)
= R (x) ,
(3.1)
is subjected to a variable change of the form, y(x) = H(x)z(t), with t = f(x), f is real valued of class C4(X), f' (x) never vanishes, H is n x n complex valued of ~ C4 (X), and H is nonsing;D:'ar ~ X,
(3.2)
then a natural canonical form for fourth order formally symmetric operators evolves. A sufficiently general form for variable change purposes is 2
L[y]
=
l:
.
(-l)~
i=O
~
A(i,j)y(j)}(i)
(3.3)
j=O
where each coefficient A(i,j) (x) on a real interval X such that
is an
A*(i,j) = A(j,i),
i,j
n x n =
complex matrix valued function
0,1,2.
(3.4)
If the indices of summation in (3.3) are allowed to run to m, a rather general 2mth order formally symmetric quasidifferential operator is obtained. If the coefficient A(m,m) is zero, then the operator is of odd order and the transformation theory for those cases can be obtained as a special case of the theory for the even order case. The discussion will be restricted to the fourth order case since it is typical of the higher order cases. 4.
KUMMER-LIOUVILLE TRANSFORMATIONS
In order to fix the setting, let efficients in (3.3). (H)
Ci(X), ~,J = 0,1,2, and the matrix i,j = 0,1,2, is hermitlan-.- - - 4 is in the domain of L if Y is of class C (X). Set
y
Suppo-6e hypothrv..,u, when app,Ue.d:to L[y]
THEOREM
L [z]
o
assume the following hypothesis on the co-
A(i,j) is of class A(x) = (A(i-;j)(Xj),
Suppose that (3.2)
US
=
T
=
f(X).
(H) ho.e.dJ... The KwnmeJt-UouvLUe vaJUable c.hange 06 (3.3) geneJta.trv.. an opeJta.tOIt LO 06 the 601tm
{(P z" + Q Zl)" 22
(Q*z" + P z' + Q z)' + (Q*Z' + P z)} 211 10
(4.1)
3
TRANSFORMA nONS OF ORDINAR Y DIFFERENTIAL OPERA TORS
.6Uch ;tW ;the identity (2.8) hold6. FuM:heJunolLe, ;the P. a.Ytd Ci(T) wdh P. heJrm.(;tianand Q. 6k.ewheJun);t[an. Aub 1
Q i
Me.
06 c..R.a.M
1
3 P (t) = {!f'1 H*A(2,2)H}(X) 2
(4.2)
and poet) = {(l/lf'l) (1/2) (H*L[H] + (L[H])*H)}(x)
(4.3)
An algorithm which yields the remaining coefficients is provided by our constructive proof. In general, the remaining coefficients are quite complicated. However, we now list several "elementary" examples.
SUppo.H. A(i,j) = 0, -<-6 i ~ j, -<-.e. A -<.6 black. d-tag a na..t. Then change 06 -<-ndependen;t valUable "p1te6e1tVe6 ;the 6oltm" in the 6eMe ;tha.;t Q1 and Q a./te zelto. Indeed, c.ho-<-ce 06 H In' t = f(x) c.a./t/t-te6 2 EXAMPLE 1.
L[y] = (R (X)y")" 2
(R (X)y')' + RO(x)y 1
(4.4)
-<-n;to (4.5)
(4.6) 60IL
r
= f'I.
Introduce the notation
+ M*l
herm(M) = (1/2) (M for any square matrix -skew(M). EXAMPLE 2.
COM-<-delt
M.
and
Observe that
L[y] = (R(x)y")".
K = RH',
S
1
skew(M) = (1/2) (M - M*) herm(M*) = herm(M)
Inttwduc.e ;the no;ta-tion
(4.7)
M2 = H*RH,
M1 = 2 herm{K*H' - H*K' - H*RH"},
poet)
LO
skew (M*)
= skew{K*H" - H* (RH") ,},
S2 ='H*K - K*H,
Then ;the c.oe6Muen;t6 -<-11.
and
r =
f'I • n
a./te g-<-ven expUcLtiy by
herm({(l/lf'I)H*L[H]}(x», { (f' / If'
I ) [- (Mi')' + M1r]
{ If' I 3H*RH} (x) ,
} (x) ,
(4.8)
{(f'/I f'l )Sl} (x), In the case of n = 1, the Pi are real valued and the Qi are pure imaginaries. Hence if the original operator had real coefficients and the variable change is real, then Q1 = Q2 B 0, (also see [2]). For scalar operators of order 2m, Reid considered operators of the type generated by the above Theorem [9, p. 168]. 5.
LINEAR CHANGE OF DEPENDENT VARIABLES
Let xl be an interior point of X and suppose that Xl = [a,b] is a compact subinterval of X such that Xl is interior to Xl' Let h be an n x 1 complex matrix valued function of class C"[a,b] with double zeros at a and b. (Note that h plays the role of a test function in the theory of distributions or the role of an admissible variation in the Calculus of Variations.) In order to motivate the choice of the general form (3.3), consider the operator L
4
C.D. AHLBRANDT et al.
of (3.1). (A more skeptical approach would be to start with form two integrations by parts to obtain (L[y] ,h)
=
Jx (L[y] ,h)dx = Ix 1
L[y]
y
(Ry",h")dx.
(4)
•
)
Per-
(5.1)
1
The effect upon L of a variable change (3.2) is determined by making the corresponding variable change on the sesquilinear form S[y,h]
Ix
=
(5.2)
(Ry",h")dx. 1
The computations are simplified if we first make a linear change of dependent variables y(x)
H(x)u(x)
=
and
hex)
=
(5.3)
H(x)v(x).
Application of Leibnitz' rule for differentiation gives S[y,h] = (C ,i
(.)
(B(i,j)uJ,v ~ )}dx
(5.4)
are the binomial coefficients) B(i,j) = (C ,)(C ,)(H(2-i»*RH(2-j). (5.5) 2 ,~ 2 ,J Note that (B(i,j»* = B(j,i), for i,j = 0,1,2. Therefore, the matrix B = (B(i,j», i,j = 0,1,2, (having n x n block entries) is hermitian on X. It follows from (5.4) that the form of L should be taken as the general form (3.3) in order to be preserved under coordinate changes. 2
Now consider
and
(. )
~
~,J=O
1
for
2
Ix {,
L
C ,j 2
of the general form (3.3) under hypothesis (H).
SA[y,h]
=
Ix {. ~
(A(i,j)y(j) ,h(i»} dx.
Set (5.6)
~,J=O
1
Then
Ix
(5.7)
(L[y] ,h)dx. 1
Introduce the notation
y=
[~:]
0 G =
[:,
H"
...
......
and similarly associate u, h, and v with SA in (5.6) may be concisely expressed by SA[y,h]
= Ix
......
(Ay,h)dx.
H 2H' u,
~l
(5.8)
Hj h,
and
v.
Then the form of (5.9)
1
The variable change (5.3) gives -+ y = Gu,
-+
-+ -+ h = Gv.
(5.10)
Replace (5.5) by B
G*AG
(5.H)
and set
IX
-+ ...
(Bu,v)dx
(5.12)
1
for the identity (5.13) under (5.3). Note that B satisfies hypothesis (H). Since v has double zeros at a and b, we may integrate certain terms by parts without destroying the form. We first modify the terms associated with the last row and column of B. The proof is facilitated by the following diagram.
TRANSFORMATIONS OF ORDINAR Y DIFFERENTIAL OPERATORS
FIGURE 1
u'
u
u" B(0,2)
v
i
v' v"
B(1,2)
B(2'O)~
Integration of
(I)
T(III~
(I)
B(2,1)
(B(2,0)u,v')'
B (2,2)
gives
Ix (B(2,0)u,v")dx = 1 Ix {-(B'(2,0)u,v') -
(5.14) (B(2,0)u',v')ldx.
1 (II)
Integration of
(B(0,2)u',v)'
gives
Ix (B(O,2)u",v)dx = 1 Ix {-(B' (0,2)u' ,v) -
(5.15) (B(0,2)u' ,v')}dx.
1 (III)
Add the expression, (whose value is (-1/2)f
to
SB[u,v].
x1
0),
([B(2,1) + B(1,2)]u',v')'dx
(5.16)
This expression expands to
o~
I Xl{«-1/2)[B'(2,1)
+ B'(1,2)]u',v')
+ «-1/2)[B(2,1) + B(1,2)]u",v')
(5.17)
+ «-1/2)[B(2,1)+ B(1,2)]u',v")}dx. (Step (III) entails removal of the hermitian parts of Define a hermitian matrix C by
Then
C(2,1)
C(O,O)
B(O,O),
C(l,O)
B(l,O) - a'(2,0),
C(2,2) = a(2,2),
C(l,l)
a(l,l) - herm[B' (2,1) + 2B(2,0) 1.
is skew'hermitian, Sa[u,vl
Repeat argument (III) on the (IV)
B(2,1)
C(2,0) =
and
B(1,2).)
° (5.18)
C(2,1) = skew(a(2,1»
satisfies hypothesis (H), and
C
= Sc[u,v].
C(l,O)
and
(5.19)
C(O,l)
terms:
Add the expression
x ([C(O,l) + C(l,O)]u,v)'dx ex~ression and define a hermitian
(5.20)
(-1/2)J
to
Sc[u,vl.
Expand the
if
i
D(i,j)
C(i,j)
or
D{l,l)
C(l,l),
D(O,O)
e(O,D) - herm(C'(l,O».
j
is
matrix
D
by
2,
(5.21)
D{1,O) = skew(C(l,O»,
Then D is tridiagonal with off diagonal blocks skew hermitian, hypothesis (H), and
D
satisfies (5.22)
6
6.
G.D. AHLBRANDT et al.
MONOTONE CHANGE OF INDEPENDENT VARIABLE
In order to complete the Kummer-Liouville variable change (3.2), set u(x)
z(t),
v(x)=k(t),
t=f(x).
(6.1)
F(x)k(t)
(6.2)
Then -+
-+
F(x)z(t)
u(x)
-+
and
vex)
for
o
[~
F =
f'I fnI
(6.3)
Substitution of (6.2) and change of variable of integration gives SD[u,vl
= fT
SE[z,kl
-+ ...
(6.4)
(Ez,k)dt, 1
(6.5)
E(t) Introduce the notation Mi (x) = D(i,i) (x), Then the given by
Mi
Sl (x) = D(l,O) (x),
are hermitian, the E(O,O)
(1/If'I)MO'
E(l,l)
jf'jM
E(2,1)
S.
1.
E(l,O) = (f'/lf' j )5 , 1
+ [(f n )2 / jf'jlM
[(f')~/jf'jlS2
+
(6.6)
S2(x) = D(2,1) (x).
are skew hermitian, and the
jf'jf~M2'
E(2,2)
E(i,j)(t) are
E(2,O) = 0
~
(6.7)
jf'j3M2
Remove the hermitian parts of E(2,l) and E(I,2) by step (III). (Note that we are now integrating with respect to t and M2 and f are functions of x.) Add the expression, whose value is zero,
-f to
SE
{([herm(E(2,I)lz',k')'}dt
(6.8)
Tl
to obtain (6.9)
SE[z,kl where
for
r
AO(i,j)
E(i,j),
A (2,1) O AO(l,l)
skew (E (2,1»,
= f'I.
if
or
E (2, 2),
(6.10)
{(f'/jf'j)[-(Mi')' + M/l}(x) Set Pi = AO(i,i),
7.
i
THE IDENTITY BE'IWEEN
L
Q2 AND
~
A (2,l), O
Q
1
= AO(l,O).
(6.11)
LO
It has been established thus far that (7.1)
(L[yl,h)X 1
At each step in the proof the coefficient matrix satisfies hypothesis (H). gration by parts in the latter expression yields (L[yl,h)X
= (Lo[zl,k)T • 1 1 Change of variable of integration gives, for ~ the inverse function to result
Inte(7.2)
f, the
TRANSFORMA TIONS OF ORDINAR Y DIFFERENTIAL OPERA TORS
{L[y],hl
7
= ({(I/!f'!)H*L[y]}(g), klT I I and hence the variational identity
(7.3)
x
(7.4) = 0 I for all test functions k. Consequently, the identity (2.8) holds at every interior point Xl of x. A limit argument establishes the identity at any boundary points of X which belong to x. (LO[Z] - {(l/!f'!lH*L[yj}(gl,kl
T
The form of the coefficient Po is easily obtained from the identity (2.8) by choosing Z to be a constant vector. Then (7.5) and by taking the transpose conjugate of both sides i t follows from skew hermitian that Qi(t) + PO(t) = {(l/!f'!)(L[H])*H}(x). Addition of (7.5) and (7.6) gives
Q l
being (7.6)
PO'
REFERENCES: [1]
Calvin D. Ahlbrandt, Equivalence of differential operators, in Proc. 1980 Dundee Conference, Lecture Notes in Math., Springer-Verlag, (in press) .
[2]
Calvin D. Ahlbrandt, Don B. Hinton, and Roger T. Lewis, The effect of variable change on oscillation and disconjugacy criteria with applications to spectral theory and asymptotic theory, J. Math. Anal. Appl., (in press).
(3]
, Transformations of second order ordinary and partial differential operators, submitted. Inversion in the unit sphere for powers of the Laplacian,
[4] submitted. (5]
F. V. Atkinson, "Discrete and Continuous Boundary Value Problems", Academic Press, New York, 1964.
[6]
Oskar Bolza, "Lectures on the Calculus of Variations", University of Chicago Press, Chicago, 1904.
[7]
W. N. Everitt, On the transformation theory of ordinary second-order linear symmetric differential equations. (preprint)
(8]
W. T. Reid, Oscillation criteria for linear differential systems with complex coefficients, Pacific J. Math. 6(1956), 733-751.
(9]
Principal solutions of non-oscillatory self-adjoint linear differential systems, Pacific J. Math. 8(1958), 147-169.
*
Partially supported by NSF grant number MeS-800S811.
This page intentionally left blank
This Page Intentionally Left Blank
Spectral Theory of Differential Operators I.W Knowles and R. T. Lewis (eds.) © North-Holland Publishing Company, 1981
FINITENESS CRITERIA FOR THE NEGATIVE SPECTRUM AND NONOSCILLATION THEORY FOR A CLASS OF HIGHER ORDER ELLIPTIC OPERATORS W. Allegretto Department of Mathematics University of Alberta Edmonton, Alberta Canada, T6G 2Gl
We extend to a class of higher order equations recent results connecting the nonoscillation of second order equations and the finiteness of the negative spectrum of associated operators. The procedure used involves the introduction of suitable second order equations and implies the localization near infinity of our considerations. The results are illustrated by considering the fourth order case and by a comparison with previous results along the same lines. INTRODUCTION Let
x = Cxl, ... ,x ) denote a point of Euclidean n-space En, 11 ~ 3, and set n for i = 1, ... ,n. We consider in an unbounded domain G, with smooth
Di = 3j3x
i
boundary, the elliptic operator
where
/::,
£
denotes the Laplacean.
with domain
We assume that
CooCG) o
q
and expression:
is real and in
2
L£oc CG ) , and that £ admits a Friedrichs extension X. It is our aim to use nonoscillation theory to obtain conditions which ensure that a_CX) , the negative spectrum of X, is finite. We remark that several other methods have been employed to guarantee the finiteness of a_CX). We refer the reader to the books of Schechter, [14], and Reed and Simon, [12], where further references may be found.
We next introduce· the form B naturally associated with 9. and term B nonsuch that if oscillatory Cat 00) iff there exists a neighbourhood N of peN n G, P a bounded domain, then there exists a constant K = KCP) > a for B(~,~) > K(~,~)
which
for all
~
€
Coo(P). o
This is an adaptation of the defini-
tion introduced by Glazman, [5]. A summary (up to 1973) of conditions for B or 9. to be nonoscillatory or oscillatory can be found in the books of Swanson, [IS], and Kreith, [7]. For more recent criteria and extensions to more general cases and to related problems, we refer to the results of MUller-Pfeiffer, [10]; Kusano and Yoshida, [8]; Hinton and Lewis, [6], and the references mentioned therein. THE CASE
m= 1
We consider first the case where £ is the second order expression: 9.~ = -/::,~ - q~. Suppose that q is regular "in bands". That is: there are smooth surfaces (i)
q
€
{Rk}~=O' tending to
Cl[Mk n G] n Loo(N
k
n G)
00, such that:
where
are neighbourhoods of 9
10
W, ALLEGR1:'TTO
Rk n G respectively; CE)
q
Ciii)
q
(iv)
L'" Q,oc in a neighbourhood of dG; is of class Ln/2 in any bounded subdomain of
is of class +
the domain bounded by
R , R (p
>
q)
and
G
G;
can be expressed as
p q u Z G with \l (Z ) and G a domain such that if pq pq pq = pq / then q E Lr 2 CT ) with r = rCT) > n.
°
T
cc G
pq
The above assumptions are a particular case of the ones introduced in [1], [2]. The following theorem is a consequence of the results established in [2]. Theorem 1. is finite.
Let
q
be regular "in bands".
Then
B is nonoscillatory iff a _ (1)
Related results have been established by Piepenbrink, [11]. and Moss and Peipenbrink, [9]. We remark that Theorem 1 remains valid if in the expression for Q, we substitute - I O. (a .. D.q,) for -6q" as long as the a., are 1
1J J
1)
reasonably regular (see [2]). We assume in the sequel, without further mention, that at least the above conditions hold on q. THE CASE
m
>
1
Serious difficulties appear to arise when an attempt is made to extend the arguments of Theorem 1 to the case m > 1. Indeed, it does not appear known in this case if the finiteness of o_CL) follows from the nonoscillation of B. We show, however, that if B is nonoscillatory by iteration of second order arguments then o_CL) is finite. Let Q denote the subset Cwo'" .,wm_l ) belongs to
of [C'" (G)] m with positive components such that Q iff Wa _ 1 and the forms:
BkCq"q,) =
n
J
W
G are nonoscillatory for
k
k = 0, ... ,m-2.
I
i=l
Note that
can always be chosen near infinity of type constants A,a,S. Theorem 2.
Let
(D q,)2 - w + q,2 i k l
Q is not empty, since the wk
Alxla(log Ixl)S
and suppose that the form
f G
B'
with suitable given by:
w _ ~ (Oi.)2 _ q.2 m l 1
is also nonoscillatory. Then there exists a finite number of linear functionals p '" such that l' f q, E P~ {Null space f } then BC.,q,) > o. (fi}i=l on Co(G) i Proof.
We express
B in the form: B(q,) =
I
I I B. 2 m j=2I a i l=l J-
(oaj+ ... +amq,)!
+ B' C.)
m~i.::j
where
a. )
is a nultinomial and
B(
BCq"
Since constant coefficient
11
FINITENESS CRITERIA AND NONOSCILLATION
differential operators map
E
J A
0
to itself, we can apply Theorem 1 to each
2 L (G)
and
Ia i I
<
B.
CL
B' , and construct the functionals
and (.
Coo(G)
f.
1
of type
f(¢)
= (D lq"E;.) )
)
with
m - l.
standard matrix and approximation argument then gives:
Corollary 3.
0_
consists of no more than
(X)
p
negati ve eigenvalues.
We note that the criteria for second order nonoscillation are now applicable, but we do not pursue this here. Instead, we consider a couple of special examples to illustrate the above results. Example 1.
Let
m
= 2,
i.e.
~~
~2q, _ qq,.
The above procedure leads to the
nonoscillation of the pair:
wi th
Wo
0, to be chosen.
>
- M - woq,
o
Wi (woD i ¢) - q¢
o
If we follow a procedure introduced for a different 2 w = div P - IpI > 0 and
problem by Protter, [13], we find that we may choose: -1
2
q:. div s - Wo lsi, where s.
P
=
(PI,···,P n ), S
=
0
(SI"",Sn)
and
Pi'
Let us further assume that G is an excerior domain. One choice 1 of P gives W (n_2)2 4- l lx l-2 (near 00). If q is specialized to be of o type alxl- 4 near then, by this method, we obtain a = (n_2)2(n_4)2 4 -2. In this special case the "optimal" a is known to be n 2 (n_4)2 4-2, for n > 4, and E
COO (G).
was obtained, [3], by nonoscillation theory, separation of variables, and estimates which depend strongly on the nature of the specific problem considered. It is interesting to note that the above "optimal" value of a is also exactly where B changes from oscillation to nonoscillation. In the above case, our method gives a worse result then what was previously known. To give a simple example of a result which does not seem obtainable by other methods we state: Example 2. Let n the cone x3 = alxl if
Ix I
lead to
<
R.
=
3, m = 2. Suppose that for some R > 0, 3G is described by (a near 1) if Ixl > R, while 3G is essentially arbitrary
Then the above arguments together with some related estimates, [4], being finite if qlxl 4 :. (9_a)2(l_a)-2 4 -2 near
o_(l)
D ¢, D ¢ as independent i j functions and appears to "change" the side boundary conditions (heuristically, from u = dU/dn = 0 to u = ~u = 0 if m = 2). It would be desirable to remove these shortcomings, but it is not clear how this can be accomplished in general.
We conclude by remarking that the method treats
Finally, we note that the localization procedures which we have introduced imply that operators with a singularity at a finite point of the boundary and/or multiple singularities can be handled in the same way, at least formally. While we do not pursue this point, we note that it may be very difficult to obtain explicit nonoscillation criteria for the above cases unless the geometry of the problem is simple near the singular set.
W.ALLECRETTO
12
REFERENCES [1]
Allegretto, W., Positive solutions and spectral properties of second order elliptic operators, Pacific J. Math., to appear.
[2]
Allegretto, W., Positive solutions of elliptic operators in unbounded domains, J. Math. Anal. Appl., to appear.
13]
Allegretto, W., Finiteness of lower spectra of a class of higher order elliptic operators, Pacific J. Math. 83 (1979) 303-309.
[4]
Allegretto, W., Nonoscillation criteria for elliptic equations in conical domains, Proc. Amer. Math. Soc. 63 (1977) 245-250.
[5]
Glazman, I.M., Direct methods of qualitative spectral analysis of singular differential operators, Israel Program for Scientific Translations (Davey and Co., New York 1965).
[6]
Hinton, D. and Lewis, R., Oscillation theory for generalized second-order differential equations, Rocky Mountain J. Math. 10 (1980) 751-766.
[7]
Kreith, K., Oscillation theory (Lecture Notes in Mathematics, Vol. 324, Springer-Verlag, Berlin 1973).
[8]
Kusano, T. and Yoshida, N., Nonlinear oscillation criteria for singular elliptic differential operators, Funkcial. Ekvac. 23 (1980) 135-142.
[9]
Moss, W. and Piepenbrink, J., Positive solutions of elliptic equations, Pacific J. Math. 75 (1978) 219-226.
[10]
Muller-Pfeiffer, E., Ein oszillationssatz fur elliptische differential gleichungen hoherer ardnung, Math. Nachr. 97 (1980) 197-202.
[11]
Piepenbrink, J., A conjecture of Glazman, J. Differential Equations 24 (1977) 173-177.
[12]
Reed, M. and Simon, B., Analysis of operators (Academic Press, New York, 1978).
[13]
Protter, M.H., Lower bounds for the first eigenvalue of elliptic equations, Annals of Math. 71 (1960) 423-444.
[14]
Schechter, M., Spectra of partial differential operators (North Holland Amsterdam, 1971).
[15]
Swanson, C.A., Comparison and oscillation theory of linear differential equations (Academic Press, New York, 1968).
Spectral Theory of Differential Operators I.W. Knowles and R. T. Lewis (eds.) © North-Holland Publishing Company, 1981
A CLASS OF LIMIT-POINT CRITERIA F. V. Atkinson University of Toronto Limit-point criteria for second-order differential operators, and limit-n criteria for 2n-th order operators, generally impose a positivity condition on the coefficient of the highest derivative, and bound other coefficients relative to it. This paper deals with criteria which focus attention on the coefficient of the independent variable, and which make no requirements of positivity or reality.
1. INTRODUCTION. We are concerned here with the classification problem for differential equations of the form - (p(x)y')' + q(x)y = AY, a < x < 00 (1.1) and, to a lesser extent, with certain variations and generalizations, such as matrix equations, higher-order equations and difference equations. For (1.1), this problem goes back to the fundamental papers of H. Weyl (30, 31), who found that just two cases were to be distinguished. If one denotes by d(A) the dimension of the space of solutions of (1.1) which are of integrable square on (a, 00), these cases are: (i) the limit-circle case,
in which
(ii) the limit-point case, in which Here
=2
d( A) d( A)
<
2
for all A , for all
A
is allowed to be real or complex. Weyl assumed that I both e[a,oo) ( or even C [a, 00) ) , and that q E C[a,oo), being real-valued, and that p
A
E
(1. 2) p (x) 00. > 0, a < x < The terms "limit-circle", "limit-point" can then be validated in terms of. the behaviour of certain circles in the complex plane; however the classification is sound without any reality or positivity hypotheses on p , q .
Weyl also proved, among much else that in the real case d (A) > I if 1m A I O. These discoveries provided the prime examples of the theory of deficiency indices of linear operators and of their extensions, and the main impetus for the extensive development of this theory in the context of differential operators in recent years. Generalization of the theory beyond Weyl's hypotheses, that p is real, positive and continuous, and q real and continuous, may be seen as a staged process. Customary assumptions are now that p(x)
>
O.
P
-1
, q
(1. J)
E
i.e. are in L(a, b) for every b E (a,oo). This allows p(x) to vanish or become infinite for individual x-values, giving rise to situations which fall outside the scope of the usual existence 13
F. V. ATKINSON
14
and uniqueness theorems. One may avoid any difficulties in this connection by interpreting (py') as a "quasi-derivative" or, better, by going from (1.1) to a first-order system. This is to be accomplished in a known manner by setting py' = z, P -1 = r , (1 .4) so that (1.1) can be replaced by y' = rz,
z' = (q - A ) y.
Here the coefficients on the right are in L (a,oo), and the solutions y, z will be continuous, indtga locally absolutely continuous functions; z will be well defined, even at points where p, y' fail to be so. A possible generalization which emerges at this point is (as suggested by Everitt) to permit p(x) to change sign, remaining real-valued, along with q(x) . Of course, we must make the restriction that l/p(x) should remain integrable at this point of change of sign. Here again the systems formulation (1.4-5) seems to allow a more natural formulation, in that r(x) can not only change sign, but also can vanish over intervals. As may be see'1 from (1), Chapter 8), much of the standard theory, including the nesting-circle phenomenon, carries over to this case; the assumption made there that r(x) > 0 was needed mainly for the semi-boundedness of the spectrum. It would seem that the detailed analysis of this case presents a considerable challenge. In another direction, one dispenses with the hypothesis that q is real-valued, and perhaps also for p ,so that there is no question of formal self-adjointness, but still retains a positivity hypothesis for p, either for p itself or for its real part if p is complex. This has been extensively investigated recently by Knowles and Race (20, 25) and others. The emphasis of the present paper will be on criteria for the presence of a non-integrable-square solution, when no hypotheses are made concerning the reality or positivity of the coefficients. The criteria will involve mainly bounds, pointwise or integral, placed on q rather than on p . These criteria will appear as special cases of general, necessary and sufficient conditions for the existence of such a solution. One approach, though a restrictive one, to the determination of d( A) is that of asymptotic integration. Subject to various restrictions, involving among other things differentiability conditions on p and q , one can approximate to the solutions, and so test their square-integrability directly.; slightly less restrictively, one may sometimes be able to test square-integrabil-ity by investigating the behaviour of suitable energy-type, or Lyapunov (or Kupcov) functions. In the reverse direction, it can be said that the proofs of weaker and more general limit-point criteria can be adapted to yield quantitative information on, so to speak, the non-square-integrability of solutions. Results of this nature were given in (4), and we shall include some here. 2. A SELECTION OF CRITERIA FOR THE CASE
P
>
0, OR
Re p
>
O.
For the sake of comparison we review some of these briefly; we start with two rather classical sufficient criteria for the limitpoint case, when p is real and positive and q real. The first is simpler, and is among the original results of Weyl (Jl). I. q(x) is bounded below on (a,oo).
A CLASS OF LIMIT-POINT CRITHRIA
15
The proof is immediate from the observation that if in (1.1) - q(X) < 0 for large X, then there is a solution which is ultimately positive and increasing, and so not s'1uare-integrable. \
Partially overlapping with this, but allowing q to become large and negative, is the criterion of Levinson (4, 21). II. There should exist a positive, locally absolutely continuous function W such that q > - W , and 2 3 1 sup pW' W< 00, J (pW)-2"dx = (2.1-2) 00
The criterion II includes I by taking not being in L{a,oo).
W
=
1
1 , subject to
p 2"
These criteria are of the global type, and have the feature that q is bounded on one side, as is of course p. In the case of II we have the hypothesis of the existence of an auxiliary function W , linked with p and q by inequalities. Subsequent developments involve restricting p and q on a sequence of intervals only, or allowing them to take complex values, or the introduction of a greater number of auxiliary functions, or again the use of integral rather than pointwise bounds on p and q. We illustrate these points in the following examples. III. The Levinson criterion II retains its validity if q is allowed to be complex, with q ~ - W being replaced by Re q ~ - W, other conditions remaining unchanged (3). The imaginary part of q plays no part in the criterion, and is arbitrary. With q complex, we can no longer speak strictly of the "limit-point case", but have rather a sufficient condition for the "J-selfadjointness" of certain operators. See (25) for more details. Going back to the real case, we have the criterion IV. We take p = 1, and assume that q(x) has a fixed lower bound on a sequence of intervals of fixed positive length, with disjoint interiors. This forms a very special case of results of Hartman, since developed further by Eastham and others (see (5»; one can also adapt criterion III so as to cover this condition (see (3). The criterion shows that the limit-point case can remain quite unaffected if p, q are left arbitrary on large parts of the axis. Moving on to the case that p may also be complex, we cite the following interval-type criterion which, though not quite the most general available, is reasonably simple, involves no auxiliary functions, but rather a choice of intervals and parameters. See (2). V. On a sequence of non-overlapping intervals (a, b ), let m m (i) Re pet) > Mm > 0, ip(t)i ~ KMm' am < t < b m (2·3) B
< a < S < b , ( 2.4) (ii) (bm - am) f Re q dt > - Km m ill a and ( iii) I (bm - am) 2/Mm = (2.5) m In this result, pet) must lie in a certain fixed sector in the right half-plane, Re q(t) satisfies a one-sided integral bound, and a sum (2.5) must be infinite (just as the integral (2.2) must be infinite). These features, or slight modifications of them, can be recognised in almost all existing limit-point criteria. 00 •
Interval-type conditions can be brought within the scope of global conditions by employing an auxiliary function which vanishes
16
F.V. ATKINSON
outside the intervals. For further developments we cite the papers of Knowles and Race (20). Read (28) and Frentzen (9). together with the survey article (8) and monograph (18). We pass now to the distinct type of limit-point criterion which forms our main concern here. 3. INTEGRAL-TYPE LIMIT-POINT CRITERIA ON q. The arguments in the sequel are largely suggested by t~e remarkable observation that (1.1) has a solution not in L (a. oo ) if
(3.1) L2 (a,oo). Here p. q may be complex-valued. and p is quite arbitrary. subject to our general conditions P -1 • q E Lloc(a,oo). (3.2 ) The above assertion. in the case p = 1 and q real. is due to Hartman (12). in whose paper it appears as a special case of the criterion q E LS(a,oo), for some s :: 1; this in turn is linked with the non-oscillatory character of (1.1) in this case when A < O. The criterion (3. i) for limit-point. with p = 1 and q real. is sO:,letimes attributed to Putnam (24). whose contribution was. however. to elucidate the nature of the spectrum in this case; I am indebted to Professor H. Kalf for clarification on this matter. A short proof of (3.1) as a limit-point criterion. with arbitrary p • is given in (18). It does not appear that there is any limit-point criterion which restricts p only. leaving q arbitrary. There have been a number of developments regarding the criterion (3.1). It has been shown by Zettl (33) that (3.1) ensures the existence of a non-integrable-square solution in the case of a class of higher-order equations (with q being still the coefficient of the dependent variable). For a slightly narrower class of higher-order eauations. Hinton (15) has given the more general criterion JT I 2 I q (tl dt = OCT), as T .... (3.3) q
E
00,
o
as sufficient for a certain bound on the dimension of the set of of L -solutions; he has also extended the result to solutions in other Lebesgue classes. In this section we go back to the second-order case - (py')' + qy = O. a 2. x < 3 . 4) with finite a • and obtain a criterion which is slightly more general than (3.3). and of course than (3.1). We also obtain a quantitative estimate of the "non-square-integrability". Theorem 1. Let -1 2 P E Lloc(a,oo), q E L loc(a,oo). (3·5) 00 ,
are solutions of (3.4) satisfying v(pu') - u(pv') = 1. we have, for some C • and writing 2 2 w = lul + Iv1 , x 2 x t 2 J w dt :: {C + 2 J (C + J Iql ds)-l dt}Yz - C . a a a In particular, (3.4) has a solution not in L 2 (a,oo) Then. if
(
u. v
(3·6) (3.7) (3.8) if
17
A CLASS OF LIMIT-POINT CRITERIA
x
2 Igl dt}-l ~ a In particular, the conclusion holds if {l +
T
J
J
2 Igl dt
=
L(a,oo).
(3.10 )
OtT log T), as T ~ 00,
a
or again if
q(x)
=
1
0(log2x ).
(J.ll)
Proof of Theorem 1. This consists of a slight development of the method used to Justify the criterion (J.l). From (J.4), (J.6) we deduce that x x 1 = v(x) {(pu') (a) + J gu dt} - u(x) {(pv') (a) + J gv dt} . (3.12) a a Hence, if 2 2 k (J.1J) C = {I (pu' ) (a) I + \ (pv' ) (a) \ } 2 , we have from (3.12) that X J.:. k (J.14) 1 < Cw 2 (x) +W2(X) J I g (t) IwYz (t) d t a x x k (3.15) < w 2 (x){C + (J Igl2 dt J w dt)Yz}. a a Squaring, we deduce that x x 2 1 :: w(x) {C + J w dtj{C + (3.16) J Igl dt}. a a Dividing by the last factor and integrating we obtain t
x
J
{C +
J
Igl2 ds}-l dt
a a from which (J.8) follows easily.
x
<
J
C
w dt +
%{J
a
x
w dt}2 , (J.17)
a
It is immediate that (J.10) implies (J.9); we note that (J.10) generalizes (J.J). We could improve (J.10) by inserting iterated logarithmic factors on the right. It is obvious that the pointwise bound (J.ll) is sufficient for (3.10); however it appears that (3.11) is not quite the best possible result of its kind. We take up this point in the next section. 4. POINTWISE LIMIT-POINT CRITERIA FOR
q .
We can obtain criteria not included in Theorem 1 by using a different treatment of (J.14), not involving the Schwarz inequality. We give the necessary argument in Lemma 1. Let A > 0, and let f, g be positive-valued functions on ~), with f locally integrable and g continuously differentlable and non-decreasing, such that x
A ::
f(x) + f(x)
Then
x
J
2
f
2
(t) dt ~ A
J f(t)g(t)dt, a x
J
(1 + 2A
t
J
a
x
=
f(x)g(x) {l +
J a
(4.1)
<
1
g(s)ds)-
a a a Proof of Lemma 1. The right of (4.1) is equal to some non-negative function hex). Multiplying by
Ag(x) + g(x)h(x)
x
<
dt.
(4.2)
A + h(x), for g(x) we have
f(t)g(t)dt},
18
F. V. ATKINSON
and so, integrating, x I (A + h(t))g(t)dt
x
J
a
a
whence
I
fg dt + ~
x
I
x
2
fg dt} ,
a
x fg dt
+ 2
[{l
a
D~fferentiating,
h
I a
(A + h)g dt }1 2 - l.
we have
f (x) g (x)
=
g (x)
(A + h(x)){l
+ 2
I
x
(A + h)g dt}-lz
a
and so f2( x)
=
(A + h(x))2 {l + 2
I
x
(A + h)g dt}-l
a >
-
A(A + h(x)) {l + 2
I
x
(A + h)g dt}-l
a
x 1 t > A I (g(t))- (g(t))(A+h(t)){l + 2 I (A+h)g ds}-l dt. a a a Here we integrate by parts, and get x 2 1 t x I f dt > lzA[(g(t))- log(l + 2 I (A + h)g ds)la + a a + lzA
I
x
g'(t)(g(t))-2 log(l + 2
J
t
(A + h)g dS) dt.
a
a
Since g' > 0, h ~ 0, the right-hand side is not increased if we replace h-by O. Doing this, and reversing the integration by parts, we obtain the required result (4.2). We now obtain a pointwise analogue of Theorem 1. Theorem 2. Let p, q satisfy (J.2), and let Iq(x) I ::. g(x) , a::. x < 00, (4.J) where g(x) is positive, non-deceasing and continuously differentiable. Let u, v be as in Theorem 1. Then, for some A E (0,=), x 2 2 2 x t I (lui + Ivl )dt ~ A I (1 + 2A I g(S)ds)-l dt. (4.4) a
a
In particular, if q(x)
=
a
O(log x), as
then (J.4) has a solution not in
x
-> 00
L2(a,00).
The bound (4.4) follows from the application of Lemma 1 to (J.14). It is immediate that if (4.5) holds, we may take g(x) to be, for large x , a multiple of log x , so that the right of (4.4) will become unbounded as x -> Again, we can improve (4.5) by inserting additional factors on the right invo.lving iterated logarithms. The bound (4.5) is, of course, an improvement of (J.ll). 00
5. DISCUSSION OF THEOREMS
1
AND
•
2
We can check the precision of Theorem 2 by means of asymptotic integration. We need the rather standard Lemma 2. Let f, g be positive-valued and continuously twice differentiable on [a,oo), and let g-
1
1
1
({2" g2)'
E BV [a, 00) ,
(
5 . 1)
19
A CLASS OF LIMIT-POINT CRITERIA
i.e. be of bounded variation over the whole semi-axis, and let also lim sup ->- 00 Then a solution
(5.2)
x
y
of ( fy' ), + gy
=
0
(5.3)
x ,
satisfies, for large
( 5.4) The proof follow0 a Kupcov-style argument, using the energy function 2.1. 2 -.1. --1.1. .1. E = Y (fg)2 + (y'£) (fg) 2 + g (f 2 g 2 )'fyy' , for which E'
We omit further details. We apply this to the example ((x 2 1og xly')' + (log x)l+c y and deduce that y(x)
=
_lo.
O(x 2(log x)
=
0,
>
0,
(5.5)
__1.._lo.s
2 2 ), 2 so that y E L (2,00). It follows that in the criterion (4.5) the power of log x cannot be increased, if this is to serve as a sufficient criterion for the limit-point case.
We can also test Theorem 2 in respect of the growth of the integral on the left of (4.4). Thus, if q(x) is bounded, we have a result of the form, for large x , x 2 2 f ( Iu I + Iv I )dt > 0 log x, (5.6) a D > O. This this is a correct order of magnitude may for some be seen in the case of the Euler equation 2
(x y' )' + Y = O. l
(5.7)
In this particular case, Theorem 1 gives (5.6) with log2 x in place of log x ; however it yields this result under the more general assumption (3.3). Whether (3.10) is in some sense best possible is not clear. However it is evident from the case of (5.5) that the power of log x on the right of (3.10) cannot be replaced by any power greater than 2 . 6. SOME VARIATIONS. We first note the adaptation of Theorem 1 to first-order two-dimensional systems of the form (1.5); it is sufficient to take the case A = O. This will permit an application to secondorder dif~erence equations. Theorem 3. Let 2 r E Lloc(a,oo), q E L loc(A,oo), (6.1) and let (3.9) hold. Then the system y' = rz , z' = qy , has a solution for which
y
is not in
(6.2) L 2 (a,oo).
For the proof we take a pair of solutions of (5.2) such that y l z2 - Y2z1 = 1 , so that
F.v. ATKINSON
20
for Some constants c l ' c 2 ' and argue as in Section J. In particular, we can use this result with the roles of interchanged, to get Theorem 4. Let -1 2 P E Lloc(a,co), q E Lloc(a, 00),
r, q (6.4)
and let
(6.5) a Then (J.4) has a solution such that py'
%
L 2 (a , 00) •
(
6.6)
For example, if p is bounded, then there is a solution such that y' is not square-integrable. In the case p = 1 this is a result of Hartman and Wintner (lJ). Still with p bounded, we can conclude from (6.6) that there is a solution such that 1
p2y'
%
L 2 (a,00),
(6.7)
(17)) of (J.4) does not exceed
so that the Dirichlet index (see
1
Next we remark that the argument of Section J can be pursued in other L-spaces; we assume (J.2) and omit detailed proofs. q
(6.8)
E L(a,oo),
then (J.4) has a solution which does not tend to zero. Theorem 6. If, as +
0,
then (J.4) has a solution not in Theorem 7.
Let a, 8
E
q
(1,00)
L(a,
00).
1, and let
satisfy lla + 1/8
E L~oc (a,oo),
(6.10) (6.11)
L(a,oo) •
a Then (J.4) has a solution not in
L 8 (a,co).
Here Theorem 7 is an extension of Theorem 1. For extensions to higher-order equations we refer to the paper of Hinton (15). Illustrating these results, we observe apropos of Theorem 5 that (xl+<5 y ') I + ~ <5 2 x- l +6 Y = 0 (6.12) has all its solutions tending to zero so that the result is precise in this check Theorem 6in the context of the (x 3 +8 y')' + (1 +J:i<5)2 x l+<5 which has all its solutions in
if <5 > 0, but not if <5 2.. 0, case. In a similar way, we can example y = 0, (6.1J)
L( 1, "')
if <5
>
0, but not if
<5
<
O.
From Theorem 7 one sees that if q is bounded there is a solution not in Lt:I (a,co) if 8 2.. 2. That this need not be so with 8 > 2 may be seen, again with the help of an Euler equation,
21
A CLASS OF LIMIT-POINT CRITERIA
namely
(6.14)
7. A GLOBAL NECESSARY AND SUFFICIENT CONDITION FOR LIMIT-POINT. As is known, in the limit-circle case the differential operator has in a certain sense a bounded inverse when its domain and range are suitably chosen, so that the prescription of unboundedness provides a sufficient condition for the limit-point case. We make this line of thought precise in the following extension of Theorem 1. We con~inue to assume the general conditions (3.2). T2(orem 8. In order that (J.4) should have a solution not in 1 a, ~it is necessary and sufficient that there be a function y with 2 y, py' E ACloc(a, 00 ), qy E L loc(a, 00 ), (7.1) and such that x (7.2) J ly2(x) 1 {I + J 1- (py')' + gyl2 dt}-l dx = 00. a
a
The necessity of this condition is trivial; we simply take to be a solution of (3.4) not of integrable square. For the sufficiency, supposing such a f
= -
y
y
to exist, we write
(py')' + qy,
(7.3)
and then have, by the variation of parameters, x x y(x) = v(x)(c + fuf dt) - u(x)(c z + J vf dt), (7.4) l a a where u, v are to satisfy (3.6). We then argue as in Section 3. Clearly, any number of sufficient criteria for the limit-point case can be obtained by choosing some y, not of integrable square, and imposing (7.2) as a requirement to be satisfied by p and q. In particular, we obtain the criterion (3.9), and so the special cases (3.10), (3.11), by choosing y = 1. We may see this as a perturbation procedure, in that we have chosen y = 1 as a solution of the base differential equation (py')' = O. We extend this remark in the context of the criterion (3.11). Theorem 9. Let (3.4) have a solution y not of integrable square, and let, for large x , 2 x 2 Iql (x) - q(x) 1 = 0 log{J Iy (t) Idt}. (7.5) Then a - (pz')' + ql Z = 0, has a solution not of integrable square. This includes the standard property that the addition of a bounded function to q does not affect the limit-point, limitcircle classification. Another result of this nature is due to Halvorsen (10), (see also (22), p. Jl). 8. NECESSARY AND SUFFICIENT CONDITIONS OF INTERVAL TYPE. It appears difficult to get sufficient conditions for the limitpoint case from Theorem 8 which have comparable scope to some of the standard ones; for example, one would like to have criteria which leave p, q arbitrary over sequences of intervals, without any form of positive requirement on p or its real part. We will therefore adapt Theorem 8 to a sequence-of-interval situa~ion. Theorem 10. In order that (3.4) have a solution not in L (a, 00) it
22
F. V ATKINSON
is necessary and sufficient that there be a sequence of intervals (a , b ), with
m
m
(8.1) and a sequence of functions y , satisfying the conditions (7.1) over the respective intervals m (am' b ), and not identically zero, m such that
(8.2) for some
m
c
with
m
I bm
a
/
Ym
/2
am:::'
dx}
%
m
c m .2. b
{Ia bm /m
(py, ') m
' and such that
m
+
I
qY,
m
/
2
dx}-%
(8.3)
We start by proving the necessity of this condition, and so assume that (3.4) has a solution y not of integrable square; the Ym will be taken to coincide, in part, with y. modified so as to satisfy (8.2) with c = a . We will arrange that each term on the left of (8.3) is ~ot l~ss than some constant, say 1 We choose
a
=
> al+l suitably large. l We take Yl = Y in (a + 1, b ), and ask in fact that b be l l l so large that the term on the left of (8.3) with m = 1 exceed 1 .The process is then to be repeated, starting with some a 2 > b l . Thus, to complete the proof of the necessity, we need only show that the specification of Yl can be completed in a suitably smooth manner.
l
a, and will take
b
If p is itself suitably smooth, we can achieve this by multiplying y by a function with is equal to 1 in (al+l, b ) l and which vani~hes together with its derivative at a . l For the general case, we write d = a + 1, and choose l l solutions u l ' vI of (3·4) such that, at d , l , , = O. pU 1, u = 0, vI = 1, pV l l l I f for the required
we set
Yl
(8.4)
- (PYl') , + qYl = fl , we shall have Yl(x)
x
x
=
vl(x)J ulfldt - ul(x)1 vlfldt a a l l f this will ensure that Yl' Py l ' -Janish at a I we take to be zero in (d , b l ), and have to choose l fl in (aI' d ) l l in such a way that y, py I at d . l We mus t therefore have 1 1 y(d l )
= I~lulfldt,
We seek
fl
-
(py')(d l )
=
(8.6)
in the form fl
=
AUI
+
BVI
'
where A, B are to be determined, and the bar indicates complex conjugation. Substituting this for fl in (8.6) we obtain a pair of equations for A, B whose determinant, a Gram matrix, is not zero since u l ' vI are linearly independent. This proves the
23
A CLASS OF LIMIT--POINT CRIThRIA
"necessity" part of Theorem 10. We turn now to the proof of the sufficiency. We denote by u, v a fixed pair of solutions of (3.4), satisfying (3.6), independent of the choice of (a, b ). With the notation (8.4) we shall have, in view of (8.2), m x m x y (x) = vex) J Ilf dt - u(x) J vf dt m c m c m m m Hence, with w as In (3.7), x !z 2 lym(x) I < w (x) \ J w I f m I dt\ ' Cm and so b b < /Ym (x) /2 w{x) J m wit) dt J m / f 2/ dt. m am am Hence
(8.7)
"
b
b 2 J m /Ym 21 dt m w dU > am am Thus the hypothesis (8.3) implies that
J
so that
b
J
m am
Ifm21
dt}-l
L(a, '" ). This completes the proof.
The condition remains necessary and sufficient under various modifications to (8.3). We could demand that the typical term on the left have a fixed positive lower bound, or again that it tend to infinity with m . If the condition be imposed with a more restrictive class of Ym ' for example those of compact support in (am' b ), the m condition will of course remain sufficient; however there seems no reason to suppose it still necessary. Test functions of this kind occur in the "singular sequence" method for locating the essential spectrum.
9. SOME SUFFICIENT CRITERIA OF INTERVAL TYPE. As with Theorem 8, so with Theorem 10 we can obtain sufficient criteria by choosing the intervals (am' b m) and functions y , and requiring that p, q satisfy (8.3). The fact that the Ym m must satisfy (8.2) implies that py' cannot vanish identically, so that p necessarily appears in ~he denominators in (8.4). Thus if there are any interval-type criteria involving q but not p , they will not be obtained by this method. To start with a simple illustration, we suppose that the (a , b) are of fixed positive length, and use the test-function m m 2 y m( x) = (x - am) ,am ~ x ~ b m . Assuming that p, p' and q are of class L2 over obtain from (8.4) the condition
tm
L{ (l pl 2 + Ip'l2 + m am as sufficient for the existence of a p, p' and q may be bounded over a length; this overlaps with Criterion result of Hartman.
IqI2}dx}-lz
= "',
2 non-L -solution. In particular, sequence of intervals of fixed IV of Section 2, an early
An alternative test-function, which does not require smooth, might be
p
to be
24
F. V. ATKINSON
This can also be used to consider cases in which rapidly oscillating.
p
is smooth but
Interval-type tests are effective even in cases where the coefficients satisfy global conditions. To illustrate this we give a result which overlaps in part with the classical Levinson result (Criterion II of Section 2) and some later developments of Hinton. Theorem 11. Let a , ~ be positive continuously differentiable functlons on [a,oo) satisfying a' = 0(1), 04>'4> -1 = 0(1), 04> -1 f/. L(a, 00'). (9.4-6) Let also p
=
o(
4> ),
p'
= o (cp a -1 ),
q
Then (3.4) has a solution not in
-2 0 (cpa ). L 2 (a,oo).
=
We now use adjoining intervals, with
bm
=
a + m l
' and take
(9.10) a l = a, a m+ l = am + a (am)' (2). We take the as in the proof of Theorem 10 in the paper now need that as in (9.1) and, writing am for a (a ) In
'
The hypotheses (9.4-5) ensure that art) / a(a m ), cp(t) / cp(a ) and their reciprocals are bounded in (a, a +1)' so that tnlr integral in (8.11) is of order cp 2 (aIn)aIn~ Th~s (8.11) will hold if 00 2 L am / cp(a ) 1
In
and, by the above remarks concerning ¢ and a , this is ensured by (9.6). This completes our sketch of the proof. In particular, we can take cp = 1, a (x) = x- l , and conclude that the conditions p ( x) = 0 ( 1) , p' (x) = 0 (x) , q (x) = 0 ( x 2 ) , (9.12) are sufficient to ensure the existence of a solution not of integrable square. The case of real positive p , and possibly complex q, is considered by Hinton (14) as a special case of a result for the 2n-th case. In some later work (see e.g. Frentzen (9)) p may be complex but lies in a sector in the right half-plane, but cannot be arbitrarily small. Since the conditions of Theorem 10 are both necessary and sufficient, it must in principle be possible to obtain from it other types of sufficient criteria, such as those which make onesided restrictions on q or its real part. It would seem that this can be done by Ym in the form vmy, where y is a solution and vm
a factor designed to bring bout (8.2) at, say, the mid-point of
(am' b m)· However the details seem to be repetitive of the known arguments for the standard tests, and will not be taken up here.
25
A CLASS OF LIMIT-POINT CRITHRIA
10. THE CASE OF A FINITE SINGULARITY. If we are considering (3.4) over a fi~ite interval (a, b), and ask whether there is a solution not in L (a, b), we can no longer derive benefit from the arguments of Sections 3 and 4. However Theorems 8 and 10 can still be used, with the obvious modifications. Thus, using Theorem 8 with b in place of ,and taking y (b - x)-t , we have that there is a solution not in L2(a, b) 00
if
p(x)
=
2
O«b - x) ),
p'(x) = O(b - x), q(x) = 0(1),
which can be checked in the case of an Euler equation. From Theorem 10 we can derive interval-type tests of a similar character. 11. THE SECOND-ORDER MATRIX CASE. We extend the above considerations to the equation - (Py')' + Qy = 0.. a < x (11.1 ) < 00 where P, Q are n-by-n matrices of functions, and y is an n-by-l column-matrix of functions. For a general formulation. we assume that P has almost everywhere an inverse R , which is locally Lebesgue integrable, as is Q; these integrability conditions are imposed in fact on the entries in these matrices. Using the quasi-derivative z = Py, we can then if necessary pass from (11.1) to the first-order system y' = Rz, for which a solution
y,
z
z' = Qy ,
(ll.2)
will be locally absolutely continuous.
We denote by I· I any convenient norm for matrices. satisfying the usual requirements. By a superscript (T) we indicate the formal transpose. We have then an almos-s complete extension of the criterion (3.9) Theorem 12. Let
IX
IQI2 dt}-l !Ja Then (11.1) and the transposed equation {l
+
- (y.p). + yQ
L(a,oo).
(ll.3)
= o.
(ll.4)
where y is a row-matrix, cannot both have more than n linearly independent solutions in L2(a,00). Here the 'term L 2 (a,00) is to be interpreted elementwise. As in the case of (11.1), we can pass from a second-order equation to a first-order system z' = yQ , y' = zR. (ll.5) with row-matrices y, z . We suppose if possible that both (11.2) and (11.5) have more than n linearly independent solutions in which y is of integrable square.
If Yl' zl form a solution of (11.2), and of (11.5), we have
Y2' z2
a solution (ll.6)
Here the left provides a non-degenerate bilinear form, with arguments in spaces of complex dimension 2n. Hence, if we have an (n+l)-dimensional space of pairs y, Z ,and likewise of Y?' z2 ' we can choose these so that th~ cotstant in (11.6) is not zero, and is for example 1 . We may thus suppose that the right of (11.6) is 1 , so that
26
F. V. ATKINSON
x
x y 2 Q d t ) y 1 ( x ) - Y2 ( x) ( z 1 ( a) + f Qy 1 d t) . a 2 a Assuming that Yl' YZ E L (a, ro), and making minor modifications in the argument of Sectlon 3. we then get a contradiction with (11.3). 1
=
(z 2 ( a) +
J
In particular, we have the conclusion that (11.1) has at most n linearly independent solutions of integrable square if (11.3) holds, and if P. Q are formally symmetric (i.e. equal to their transposes), or again if the are hermitian symmetric. Systems of somewhat more general form than (11.1), in which P enjoys some positivity property, have been considered by Frentzen (13). 12. EXTENSION TO FIRST-ORDER CANONICAL SYSTEMS. We now extend this type of reasoning to systems of the form Jy' = A(x)y ,
a'::'
x
<
(12.1)
00
under the basic assumptions: (i) J is a constant square matrix satisfying J = - JT, J2 = - 1,
(12.2- 3)
where I is the identity matrix, (ii) A(x) is a square matrix whose entries are locally Lebesgue integrable, and which satisfies A(x) = AT(x) • T
where ( )
again denotes the transpo3e.
We denote by equation and note that
Y(x) JY'
=
the solution of the corresponding matrix A(x)Y.
T
Y (x) JY ( x)
=
Y(a) = I . J
(12.4)
As was shown in Theorems 3 and 4. it may happen that more than one "non-integrable-square" property holds for a system of a given form. under appropriate hypotheses on the coefficients. We can derive these by introducing an auxiliary projector P • to have the following properties: (iii)
P
is a constant square matrix. satisfying P = pT p2 'I o.
(12.5)
and A(x)P
PA(x)
.
(12.6)
We have. with these hypotheses. and any standard norm. Theorem 13· Let x {l + f IpA(t) 12 dt} -1 jl L(a,oo). a 2 Then (12.1) has a solution such that Py jl L (a.ro). It follows from (12.2-4) that YJy T = J • and so we have PJY(x)JyT(x)P = - P . x PJY(x) fa PJY' (t)dt + PJY(a). and PJY' = PAY = PAPY. Hence
Now
- P
=
x
PJY( a) +
J
PA(t)Py(t)dt}
T
J
{(PY(xl) };
a taking norms and arguing as for (3.14) we conclude that
(12.8)
27
A CLASS Or LIMIT-·POlNT CRJ'I'ERIA
PY(x) is not square-integrable. Hence at least one column in this matrix is not square-integrable, which proves the result. We can also apply to (12.8), after taking norms, the argument of Lemma 1, as in Theorem 2. Thus, we have Theorem 14. Let PA(x) = O(g(x», (12·9) where g(x) is positive, non-decreasing, and continuously differentiable, and such that x
{I +
~
J g{t)dt}-l
L{a,oo).
a
Then (12.1) has a solution such that
Py
(12.10)
L 2 {a,oo).
~
As remarked in (4.5), it will be sufficient that PA(x) I).
APPLICATIO~
= O(log x).
(12.10)
TO THE FOURTH-ORDER SCALAR CASE.
We remark without going into details that Theorems 1 and 2 are included in the last two theorems, by suitable choice of the idempotent P , as are Theorems ) and 4. We pass on to the case of (l).l )
where P2' PI and complex-valued.
q
are locally Ll-functions, possibly
We introduce the quasi-derivatives Yl
y, Y2 = y'
, y) = Y"/P2 ' Y4 = (Y"/P2)' + PlY"
so that y , = Y2 ' Y2' = P2Y) 1
[~
0
0
0
1
-1
0
0
0
-;I
w'
[~
w
0
0
-Pl
0
0
-P2
1
0
(13.)
qY l
, Y)' = Y4 - P I Y2 ' Y4'
We can present this in canonical form. We write and then arrive at the system
(1).2)
col (Yl' .. 'Y4)
;1
(13.4)
w
This has the form (12.1), with (12.2-4) being satisfied. For a first application of the results of the last Section, we take P to be the square 4-by-4 matrix with 1 in the first row and column, and zeros elsewhere. From Theorems I) and 14 we deduce Theorem 15. Let q satisfy ().9), or in particular ().10), or else let it satisfy (4.5). Then (1).1) has a solution not of integrable square. Here P2 and PI are quite unrestricted, except for the local integrability requirement. This result is given by Hinton (18), with the slightly more restrictive condition ().). The result does not assert the "limit-2" situation, since there may be as many as three linearly independent solutions of integrable square under the assumptions of Theorem 15. Hinton (18) illustrates
28
F. V. ATKINSON
this possibility by examples based on work of Walker (29) and of Wood (32). The same phenomenon is exhibited by the Euler equation (x 5 y")" + 4(x 3 y')' = 0 , (13.5) 1, x-I, x- 2
which has as solutions
and
x-llog x .
Let next P have a "1" in the third position in the leading diagonal. and zeros elsewhere. As applied to A. the matrix on the right of (1}.4). P picks out the entry - P2 . Thus we get Theorem 16. Let P2 satisfy any of the conditions imposed on q in Theorem 15. Then (13.1) has a solution such that Y"/P2 is not of integrable square. In particular. if in addition to the hypotheses mentioned for P2 in Theorem 15. we have that 1/P2 is bounded. then the Dirichlet index of (13.1) is at most } By taking P to be an idempotent matrix with l's in the second and fourth places on the leading diagonal, we can conclude that if PI satisfies similar hypotheses, then there is a solution such that Y2 and Y4 are not both of integrable square. By writing (l}.l) in the form of a matrix Sturm-Liouville system (see (1). Section 10.6). 'Ne can deduce from Theorem 12 that if P2 and q both satisfy (3.9). then the set of solutions such both y, Y"/P2 are of integrable square has dimension at most 14. CONDITIONS FOR THE LIMIT-2 FOURTH-ORDER CASE.
2.
As noted by Hinton (15), and as illustrated by (1}.5), the boundedness of q does not, in the absence of restrictions on PI and P2 ' ensure that the space of square-integrable solutions does not have dimension greater than 2 . Hinton showed. however, that this conclusion can be drawn if PI = 0, and q satisfies (}.}); he dealt actually a rather more general problem. Confining attention to the present case of (l}.l), one may ask such questions as whether there exist conditions on q and PI which ensure that the space of square-integrable solutions has dimension less than three. whatever the choice of P2' other than the condition Pl = O. Conditions can certainly be found which limit PI in terms or q and P2 . Such a result is Theorem 17. Let p be a positive locally L 2 -function on (a. 00) such that x (1 + J p2 dt)-l."" L( a, 00 ) , (14.1) .1" a
and assume that q
=
O(
P1'(x)
=
(14.2)
p ) •
(14.})
O(xp(x»,
x
Pl(x)(l + and
J
ItP2(t) Idt)
0(1),
(14.4)
a
a .::. t
<
x.
Then the set of square-integrable solutions of (l}.l) has dimension at most 2.
29
A CLASS OF LIMIT-POINT CRITERIA
The proof follows similar lines to the investigation (18). With the notation (13.2) one has, if y is a solution and z a second solution of (13.1), the Lagrange identity y l z 4 - Y2 z 3 + Y3 z 2 - Y4z1 = const. Here the left represents a non-deg",nerate bilinear form on complex linear space of four dimensions, and so if the set of square-integrable solutions has dimension at least three, we can arrange that arrange that this form has the value 1 , for a pair of such solutions. Thus we shall have 1
,
We write
w so that W
L(a,
E
(14.8)
00).
We multiply (14.6) by (T - t) and integrate over (a, T). Using (13.3) and certain partial integrations, we get T 2 T !(T - a) = 2 J (T - t) (Yl z4 - Y4 z 1)dt + J (y3z1 a a T
+
J a
(T -
t)Pl(z2 Yl - y 2 z 1 )dt + 0(1).
We noW estimate the various terms on the right. We have x x Y 4 ( x) = Y 4 ( a) - f elY 1 d t = 0 ( 1) + 0 ( J a a and likewise for
(14.10)
z4(x). Similarly,
In the last term we integrate by parts and find, using (14.2-3), x
J
a
for large y 2 (x)
=
~
p
w2 dt +
~
Ip 1 (x) I w2 ( x) ) ,
x, and again likewise for x 0(1) + P2Y3dt ,
( 14.11)
z3' Finally, we have
!
and, using (14.4-5), we find after some calculation that x ~ Pl(x)Y2(x) = 0(1 + J pW2 dt) Collecting these esti~ates we find from (14.9) that T ~ ! (T - a ) 2 = 0 { 1 + T J #2 ( t ) (1 + a a We may re-write this in the form
(14.12)
30
F. V. ATKINSON
Yz
p w ds)dt > AT - B , a
a
(14.13)
for some A > 0; this forms a sort of integrated version of the inequality (3.14), and may be handled by a slight elaboration of the previous method. One notes first that the "1" is inessential; since w E L( a,oo) T 1 1 we have that J w2 d t = 0 (T2) . a Dropping it, at the cost of a change in A and B, and using the Schwarz inequality, we may assume that T
J
t
J
wIt) (
w ds)
for some Al .,. Write now
0
t
(14.14)
a
a
a
(j
and
some
a
a.
>
l t
Wet) = wet)
J
a
w(s)ds,
so that, by (14.8), W
E
L( a,
00
)
•
(14.15)
Then T
J a
W(t)dt
l
t 2 -1 [{J p ds} a
T 2 T W(T)J p dsdTJ a a a 1 1
t
J
by an integration by parts. Hence, by (14.14), T
J W(t)dt a
>
1
t T t [A1t{J p2dS}-lJ~ + J p2(t){J p2ds }-2 Alt dt alaI a T t Al J {J p2ds }-1 dt. a a 1
This contradicts (14.1), and so the hypothesis of more than two linearly independent square integrable solutions must be false.
The hypotheses of Theorem 17 are certainly satisfied if PI = 0, and q satisfies (J.9); in that case p? is unrestricted, apart from our general local integrability hypothesis. The condition (14.4) requires in any case that P be bounded, and may turn out to be too restrictive. We c~n however say, for example, that the conclusion of Theorem 17 will hold i f q, PI' PI' are bounded, and if
P2(x ) = O(x - 2- 0 ) , for some
0 > O.
A CLASS OF LIMIT-POINT
31
CRn~RL4
15. SECOND-ORDER DIFFERENCE EQUATIONS. We consider in this section the recurrence relation
or, in difference equation form
o ,
(15.2)
where (15. J) We are concerned with whether there is a solution not in that is to say such that
)l
2
(15.4) A recent discussion of this and allied questions is due to Hinton and Lewis (16), actually in the weighted case, when factors a are inserted in (15.4). One may attack such questions by adapt~ng the differentiation and integration arguments of the foregoing to the discrete setting, or by using the theory of first-order systems such as (6.2), or again by using the theory of integral equations with Stieltjes integrals (22). Using the first of these approaches we give an analogue of Theorem 8. We use the notation
Theorem 18. In order that (15.1) have a solution not of summable square it is necessary and sufficient that there exist a sequence {Y } such that n
(15.5) {Y } is a As with Theorem 8, the necessity is trivial; if n solution of (15.1) not of summable square, then we have (15.5) since the r m are zero.
For the sufficiency, we use the discrete version of the variation of parameters. We choose solutions of (15.1) such that
and then have 05.7) as in ((19), p.4J7), for some constants
vn = Iwn 12
+
A, B
Iz n 12 ,
we have, for some constant C , 1; n Iy n I < V n 2(C + I1
Hence, if (15.8)
k
Irm Ivm 2).
F. V. ATKINSON
32
Hence
iyn I
2 <
v
-n
n
Z
I1 I r m I ) (C
+
(C
n
+ LV), m 1
and we get the result on dividing by the first bracketed factor on the right and summing. Just as with Theorem 8, we may obtain sufficient conditions for this "limit-point" situation by choosing some sequence {y}, not of summable square, and imposing (15.5) as a condition on n the coefficients. Once more, a natural choice is Yn = 1 , and so we get Theorem 19. If
(15·10) then (15.1) has a solution not of summable square. In particular, it is sufficient that
Ib m ' I Z
1
.... '" as n to (J.IO-H). b ' n
(15·11)
0 (n log n)
=
, or of course that
b
n
'
o( log2n ),
One may conjecture that, in analogy to = O(log n) might suffice.
in analogy
(4.5), the condition
One can also formulate an "interval-type" criterion. The an' b n are now to be positive integers, such that
al
<
bl 2 a2
<
b
Z
2 '" (n)
(an' b n ) we associate a sequence Ym • m = an-I, ...• b n ' with Ym(n) = 0. for m = an-I, an . The condition is then that
With each interval
I
d:n
n
a
Iy (n) Iz}y.{In m
a
n
in analogy to (8.3); here
n
rm(n)
In particular, taking b n we get the known criterion
is defined an + 1, and
"',
as above. Ym
(n) = 1
for
(15.12)
as sufficient for the existence of a non-sunooable square solution (see (16), p. 435). As remarked there (p. 436). this criterion is quite independent of the b ; in contrast. the limit-point, limit-circle classification ofn(l.l) is certainly not independent of q if we take p = 1 . Let us now look briefly at the treatment of (15.1) by means of first-order systems. one version of which was given in «1). Chapter 8). We introduce a pair of functions u(t). vet) by the following:
33
A CLASS OF LIMIT-POINT CRITERIA
(i )
2n - 1
<
t
<
2n ,
u(t) = Yn ' u'(t) = 0 , vet) - c v ' ( t) = b n ' y n = b ' u ( t) n ( ii)
2n
u(t)
<
t
<
n-
ley
n
- y
n-
1) + (t-2n+l)b 'y , n n
,
2n + 1,
Yn + (t
vet) = cn(Y + n l
2n) (Yn+l - y n ), u' (t) Y ), n
v' (t)
Yn + l
- Yn
cn
-1
vet),
0
{Y } is equivalent n to the square integrability of the function u(t). Thus the criterion (15.10) appears as a special case of Theorem 3.
Here the square summability of the sequence
It seems likely that the criterion (15.12) could be seen as a case of an "interval-type" cri-~3rion for (6.2), as an analogue for (6.2) of Theorem 10. However such an analogue is not presently to hand. REFERENCES: (1)
Atkinson, F. V., Discrete and continuous boundary probleNs, (Academic Press, New York and London, 1964).
(2)
Atkinson, F. V., Limit-n criteria of Soc. Edin. (A) 73(1975), 167-198.
~ntegral
type, Proc. Roy.
(3)
Atkinson, F, V. and Evans, W. D., Solutions of a differential equation which are not of integrable square, Math. Z. 127, (1972), 323-332. (4) Coddington, E, A. and Levinson, N., Theory of ordinary differential equations, (McGraw-Hill, New York, 1955) (5)
Eastham, M. S. P., "On a limit-point method of Hartman, Bull London Math. Soc. 4«1972), 340-344.
(6)
Evans, W. D., On the limit-point, limit-circle classification of a second-order differential equation with a complex coefficient, J. London Math. Soc. (2), 4(1971, 245-256.
(7)
Evans, W. D., On limit-point and Dirichlet-type results for second-order differential eXpressions, in: Ordinary and Partial Differential Equations, Dundee, 1976, Lecture Notes in Mathematics, # 564, (Springer-Verlag, Berlin-HeidelbergNew York, 1976), pp. 78 - 92.
(8)
Everitt, W. N., On the deficiency differential operators 1910-1976, Proceedings from the Uppsala 1977 Differential Equations, (Uppsala,
index problem for orrtinary in: Differential Equations, International Conference on 1977), 62 - 81.
Frentzen, H., Limit-point criteria for systems of differential equations, Proc. Roy. Soc. Edin (A), 85 (1980), 233-245. (10) Halvorsen, S. G., On the quadratic integrability of solutions of x" + fx = 0, Math. Scand. 14(1964), 111-119. 2 ( 11) Hartman, P., On the number of L -solutions of x" + q(t)x =0, Amer. J. Math., 73(1971), 635-645. (12) Hartman, P., Differential equations with non-oscillatory eigenfunctions, Duke, Math. J. 15(1948), 697-709·
34
(13)
(14) (15)
F. V. ATKINSON
Hartman, P. and Wintner, A., On the derivatives of the solutions of one-dimensional wave equations, Amer. J. Math. 72(1950), 148-156. Hinton, D., Limit point criteria for differential equations, Canad. J. Math. 24(1972), 293-305. Hinton. D., Solutions of (ry(n»(n) + qy = 0 of class Lp[O,oo), Proc. Amer. Math. Soc. 32(1972), 134-138.
(16)
Hinton. D. and Lewis, R., Spectral analysis of second order difference equations, J. Math. Anal. Appl. 63(1978), 421-438.
(17)
Kauffman. R. M., The number of Dirichlet solutions to a class of linear ordinary differential equations", J. Diff. Equ. 31 (1979), 117-129· Kauffman, R. M., Read, T. T., Zettl, A., The deficiency index problem for powers of ordinary differential expressions, Lecture Notes in Mathematics, # 621, (Springer-Verlag, BerlinHeidelberg-New York, 1977). 2 Knowles, I., On the number of L -s01utions of second-order linear differential equations, Proc. Roy Soc. Edin.(A), 80(1978). 1-13. Knowles, I. and Race, D., On the point spectra of complex Sturm-Liouville operators, Proc. Roy. Soc. Edin. (A), 85(1980). 263-289.
(18)
(19)
(20)
(21)
Levinson. N., Criteria for the limit-P9int case for secondorder linear differential operators, Casopis pro pestovani matematikya fisiky, 74(1949), 17-20.
(22)
Mingarelli, A., Volterra-Stieltjes integral equations and generalized differential expressions, Ph. D. Thesis, Dept. of Math., University of Toronto, (July, 1979).
(23)
Mingarelli, A., A limit-point criterion for a three-term recurrence relation, C. R. Math. Reports Acad. Sci. Canada, (1981, to appear).
(24)
Putnam, C. R., On the spectra of certain boundary value problems, Amer. J. Math. 71(1949), 109-111. Race, D., On the location of the essential spectra and regularity fields of complex Sturm-Liouville operators, Proc. Roy. Soc. Edin. (A), 85(1980), 1-14.
(25)
(26)
Read, T. T., A limit-point criterion for expressions with oscillatory coefficients, Pacific J. Math. 66(1976), 243-255.
(27)
Read, T. T., A limit-point criterion for expressions with intermittently positive coefficients, J. London Math. Soc.(2). 15(1977), 271-276. Read, T. T., A limit-point criterion for - (py')' + qy, in: Everitt, W. N. and Sleeman, B. D. (eds.), Ordinary and Partial Differential Equations, Proc. Conf. Dundee, 1976, Lecture Notes in Mathematics, (Springer-Verlag, Berlin-Heidelberg-New York, 1976) . Walker, P., Deficiency indices of fourth-order singular differential operators, J. Diff. Equ. 9(1971), 133-140. Weyl, H., Uber gew5hnliche lineare Differentialgleichungen mit singul~ren Stellen und ihre Eigenfunktionen, GBtt. Nachr. Math.-Phys. Klasse 37-63(1909), 195-221.
(28)
(29) (30)
A CLASS OF LTMIT-POINT CRITERIA
35
()1)
Weyl, H., Uber gew5hnliche Differentialgleichungen mit Singularit~ten und die zugeh5rigen Entwicklungen willk~rlicher Funktionen, Math. Ann. 68(1910), 220-269.
()2)
Wood, A. D., Deficiency indices of some fourth-order differential operators, J. London Math. Soc.(2), )(1971), 96-100.
()))
Zettl, A., A note on square integrable solutions of linear differential equations, Proc. Amer. Math. Soc. 21(1969), 671-672.
This page intentionally left blank
This Page Intentionally Left Blank
Spectral Theory of Differential Operators I.W. Knowles and R. T. Lewis feds.! © North-Holland Publishing Company, 1981
BOUNDS FOR THE LINEARLY PERTURBED EIGENVALUE PROBLEM Michael F. Barnsley School of Mathematics Georgia Institute of Technology Atlanta, Georgia 30332
A self-adjoint family of operators of type (A), depending linearly on the perturbation parameter, is considered. Some Taylor series coefficients in the expansion(s) of one (or more) eigenvalues are supposed to be given. Theorems are presented showing how such local data can provide best possible bounds on eigenvalues of the family. The bounds apply globally in the perturbation parameter.
INTRODUCTION Let A and B be self-adjoint linear operators in a Hilbert space h. Let H = H (x) = A+xB be a self-adjoint operator for all x £ I where I is a real interval which contains the origin as an interior point. The domain of an operator C in h is denoted Dc' Then DH is assumed to be independent of x, so that DH = DA c DB' It is furthermore assumed that the lowest portion of the spectrum of H(x) is discrete, the first N eigenvalues of H(x) are expressed Al(x)
~
A2
(X)
~
•••
~
AN(X),
xEI,
these being counted according to their multiplicities. It is supposed that these eigenvalues are in fact nondegenerate except possibly at finitely many nonzero points belonging to I. Under these conditions it is well known that the functions {An(x)}:=l are regular in some neighborhood of x = 0, and that their Taylor series expansions can in principle be obtained through the Rayleigh Schrodinger perturbation equations [lJ. We will write A (xl =)~ J:..>.(j)x j n E {l,2,"',N}, n
L
J=O
J!
n
'
for the expansion about x = 0 of the nth eigenvalue. We consider the following problem. Suppose that one is given a finite and typically small set of Taylor series data; for example, sup!,'ose that one knows the numerical values of the set of coefficients S = {A (0) A(1) A(2) A(3) A (O)} Then, on the basl' s 1'1'1'1'2
.
of the given information, what are the best possible bounds which can be imposed upon the An(X) 's, for all x £ I? (For the answer to such a question when the set S above is given, see Theorem 3) Typically one is interested in the few lowest levels. The theorems presented here give a good indication of the kind of results which can be obtained. Theorem 2 is perhaps the most surprising: information relating to AK(X), where K may be unknown, yields a bound 37
38
M.F. R4RNSLEY
on "2(x). One reason for studying the above problem comes from theoretical chemistry. The Born-Oppenheimer potential energy curves {Ei (R) I i=O, 1,2, ... } are the eigenvalues of the hamil tonian
H(~,R)
N
=
{-I ., . 1 l=
N
v~
-I -l. 1 l=
Z
N
-2' '~l
a
T"X,T I~il l=
Z
sR I'+I . I i~i- ~ l<] A
ZZ
1
[x.-x.[ -l-]
+~ }, R
which corresponds to a molecular system with N electrons, and two nuclei a and S fixed at a distance R apart. The origin of coordinates is at a; ~i denotes the position vector of the ith electron; ~ = (~1'~2'···'~N); Za and Zs are the nuclear charges; and z is a unit vector along the internuclear axis. H(x,R) is assumed to be an essentially self-adjoint operator with its-domain in 2 L2( m3N ). It is readily found that R H(Rx,R) depends linearly on the parameter R. Furthermore, in the case of diatomic molecules, Taylor series data pertaining to the eigenvalues of this operator can be obtained by inversion of spectroscopic data. Coefficients in the expansions of eigenvalues of H(x,R) about equilibrium internuclear distances are available. A list of references is given in [5J. Using such data Theorems 1, 2, and 3 can be applied to provide bounds on potential energy curves. Other applications for results concerning the considered problem can be deduced from situations mentioned by Narnhofer and Thirring [4J. THEOREMS AND EXAMPLES Theorem 1. The convexity bound " (x) < ,,~O) + ,,(l)x 1
-]
]
is valid for all x E I and all j E {I, 2, ..• , N}. More generally, for any M E {1,2,···,N}, and any permutation 0 of {1,2,···,N}, M
I
".l (x)
i=l
<
M
(0)
I (" a ,') l
i=l
+
X" a(1),.l 1 )
for all x
E
I.
Each of these bounds is best possible on the basis of the information which it uses. Remark. To see the best possible nature of the bounds consider the situation when both A and B are diagonal NXN matrices, so that N
{"n(x) }n=l consists of segments of straight lines.
Such an example
is illustrated in Figure 1. In this case the convexity bound "l(x) ~ ,,~O) + ,,~l)x is saturated for x E [2,3J. Similarly one sees how saturation can occur when
r.1 >
1.
Remark. In [4J the convexity bound is given for j = 1. There also, a variety of applications to theoretical physics is considered. It seems clear from Figure 1 that much advantage can be gained with j chosen freely.
39
BOUNDS POR THE LL\'L1RLY PliR1URBliD lilGENVALUE PROHLEM
y
Figure 1 The bound A (x) < A (0)+:\ (l)x 1
-
3
3
is saturated for x E [2,3]. y
A 2 (x)
t
o
+
'.
x
(2)
Theorem 2. Let k c {1,2,"',Nj and Ak ~ O. Then real polynomials Pix) and Q(x), of degrees 1 and 2 respectively, can be chosen so that 4 Pix) Ak(X) - Q(x) = 0(x ) and P(O) = 1; moreover A2 (X)
~
Q(x)/P(x)
for all xcI such that A~2)_ ~A~3)x > O.
The bound is best possible
on the basis of the information which it uses. Remark: The function Q(x)/P(x) is the ~2/1J Pade approximant (P.A.) to Ak(x). Also, the tangent A~O)+ XA~l used in Theorem 1 is in fact the [l/OJ P.A. to Ak(x). So far, roughly speaking, we have asserted that any [l/OJ P.A. bounds Al(x) whilst any [2/1J P.A. bounds A2(x) A highly conditional assertion about the relationship between any [M/(M-l) J P.A. and AM(X) for t1 c {1,2,···,N} may be valid. Remark. For the eigenvalues of the linearly perturbed operator
~
+ x 2 + yx4} in L2, dx2 where y is the perturbation parameter, Simon et al. [2,3J have established Stieltjes characterizations which enable bounds to be imposed using P.A. 'so {_
Wilson et al [7J have demonstrated an invariance property which suggests that, from among the various rational approximants which might be used for linearly perturbed eigenvalues, the [M/(M-l) JP~}s are the most appropriate ones.
40
M.F. BARNSLEY
Example.
~) ~
Suppose we have the information ,(0)= 0
Al
A (1)= 1 1 '
'
A(2)= -1 and A(3)= -3. 1 ' 1
Then Theorem 2 yields the upper-bound marked a in Figure 2. formation
S)
, (0) _
112
-
2
, (1)
,
112
=
-1
'
1.(2)= 1, and 2
provides the bound labelled S. simultaneously valid is where
, (3)
=
The in-
3
"2
'
One system for which u) and S) are
A+xB = (x x for which A (x) is the curve marked I in the figure. 2
Figure 2 a and S are [2/1J PA upper bounds for I which denotes a 1.2 (x) .
4
2
-2
-4
x
(2)
Theorem 3 Let Al ~ O. Then real polynomials P(x) and Q(x) , of degrees 1 and 2 respectively, can be chosen so that 2
Al (x) + P (x)
4
Al (x) + Q (x) = 0 (x ),
and 2
A (x) + P(x) 2 Moreover
A (x) + Q(x) 2 for all x
= E
O(x).
I,
where A (x) denotes the lowest root of the equation 2 A (x) + P(x)
A(X) + Q(x) = 0;
and this bound is best possible on the basis of the information which it uses.
41
BOUNDS FOR THE LINEA RL Y PDR TURBDD mCDNVAL UD PROBLDM
OUTLINE PROOFS OF THE THEOREMS We use the notation <, > for the inner product in h. We write for an eigenvector such that
~.
(x)
J
(1) = A.(x)~.(x), and q.(x),l/J.(x» = 1 J J J J J for all x in some neighborhood of O. We let N denote a complex neighborhood of 0 such that, for each j, A.(X) and ~. (x) are regular J and bounded for all x EN. J k k 2k+l
L I A(P) P O( 2k+2) ( 2) n=O n. J n=O J p=O PT j x + x , for x E N, (A+xB)~.(x)
-l,
and k <
L
( 3)
n=O which are valid for all k c {0,1,2,"'} and all n E {1,2,···,N}. These equations are one way to express the basic equations of Rayleigh-Schrodinger perturbation theory. From them one discovers that the matrix elements
expresse~
(A+xB) ~ ~n) > for m and n in
in terms
together with the overlaps am n = {1,2, ... ,kL '
o~ A~P)for
p • {O,l,"
·,2k+l}
<~~m) ,~~n) > for m and n in J
]
Proof of Theorem 1. It follows from a theorem of Ky Fan [6J that
L~=l Ai(x) is the minimum of L~=l<¢i,{A+XB)¢i> when the M orthonormill. vectors ¢. (i (0)
l
¢i = ~o(i)
E
{1,2,···,M}) vary in the domain of A. Here we choose
for i
E
{1,2,"',M} and note that from (2) with k=O we
have (0)
( 0)
<~o (i) ,
(A+xB) l/Jo(i»
A (0) (i)
a
+
(1)
x\, (i)
.
Proof of Theorem 3. Let P denote the orthogonal projector corresponding to any two dimensional subspace of DA. Then the RayleighRitz variational principle provides that the two eigenvalues ~
Al(x)
~
.
~
A (X) of P(A+xB)P restrlcted to Ph obey 2 ~ ~ Al(x) $ Al (x) and A (X) < A (X) for all x E I. 2 2 ~ ~ Note that Al(x) and A (X) are the roots of 2 <¢l' (A+xB-A) ¢l>' <¢l' (A+xB-A) ¢2> 0
Det
( 4)
<¢2' (A+xB-A) ¢1>' <¢2' (A+xB-A) ¢2> where {¢1'¢2} spans Ph. Let us choose ¢ = ~(O) and ¢2 1 1
l/J (1) 1 .
Then we find
42
M.F, BARNSLEY
a+/S
'V
and \2 (x)
( 5)
20 11 '
where 1 XA (3)
[';,A (2)
6'
1
1
J
(6)
and ,(3).2 +
1
6' x Al
J
2 [A (2) J2 x all 1
( 7)
That these expressions have been successfully written in terms ,(1) (2) (3) } {-A(O) 1 ' Al ' Al ' Al 'V' 0Il alone follows from (2) and '(3). We now use the constraint A (0) ~ A~O) together with the fact that A (2) < () 2 1 (which follows from convexity), to deduce u
< _~A(2)/
11 -
2
1
(A(O)_ A(O» 2
( 8)
1
Finally, we notice that for all x
E
I,
and that the right hand side here is monotone increasing in a . The maximum possible value is achieved when the equality sign ll holds in (8). The resulting bound can be shown by direct algebra to be exactly the one claimed in the theorem. The best possible nature of the bound follows from the fact that
~l (x)
with all
=-~Ai2) / (A~O)_
AiD»
is the lowest eigenvalue of a
certain self-adjoint two-by-two matrix which depends linearly on x. The eigenvalues of this matrix match all of the information used in the construction of the bound. Proof of Theorem 2. This )roceeds much as does the proof of Theorem 3. Let us choose ¢l = ~~O and ¢2 = ~~l) in (4). Then we find that
K
~l (x) and (x) are again given by (5) but where, in the expres2 j ) is replaced by A~j) for j E {0,1,2,3] and sions (6) and (7),
;i
This time all we know about where all now refers to the kth level. 'V Upon maximizing the bound A (X) < A (X) with all is 0 < all < 2 2 respect to all we oLLi'tin the claimed bound. Note that for xj
of the result is proved by showing that there is a simple NXN matrix whose second eigenvalue, for given x, approaches the bound arbitrarily closely, and whose eigenvalues match the information used in the construction of the bound.
BOUNDS FOR THE LINEA RL Y PER TURBED EIGENVALUn PROBLEM
43
REFERENCES [lJ T. Kato, Perturbation Theory for Linear Operators, SpringerVerlag, New York, 1976. [2J B. Simon, Coupling constant analyticity for the anharmonic oscillator, Ann. of Phys. ~ (1970), 76-136. [3 I J.J. Loeffel, A. Martin, B. SimoD, A.S. Wightman, Pade approximants and the anharmonic oscilJator, Physics Letters 30B (1969), 656-658. L4J H. Narnhofer, W. Thirring, Convexity properties for coulomb systems, Acta Physica A~striaca !! (1975), 281-297. [5J M. Barnsley and J.G. Aguilar, On the approximation of potential energy functions for diatomic molecules, International JOllrnal of Quantum Chemistry 13 (1978),642-677. [6J Ky Fan, On a theorem of Weyl concerning eigenvalues of lineal transformations, Proc. Nat. Acad. Scirnces 35 (1949),652-655. [7J S. Wilson, D.M. Silver, and R.A. Farrell, Special invariance properties of the [N+l/NJ Pade approximants in RayleighSch;~odinger perturbation theory. Proc. R. Soc. Lond. A365 (1977), 363-374.
This page intentionally left blank
This Page Intentionally Left Blank
Spectral Theory of Differential Operators I. W. Knowles and R. T. Lewis reds.) © North-Holland Publishing Company, 1981
ANALYSIS OF BOLTZMANN EQUATIONS IN HILBERT SPACE BY
MEANS
OF A NON-LINEAR EIGENVALUE PROPERTY
M. F. Barnsley, J. V. Herod, D. L. Mosher, G. B. Passty School of Mathematics Georgia Institute of Technology Atlanta, Georgia 30332 By considering an appropriate realization in l2, this paper presents a study of the Tjon-Wu model for the Boltzmann equation: dy Y . _ - j dzu(t,y-z)u(t,z),u(O,xl-uO(x). 0 x y This realization is possible because of a nonlinear (l
a~ (t,x)+u(t,x)=j
eigenvalue property which the integral operation on the right side of the above equation possesses. Existence and asymptotic results are presented and a twoparameter semigroup of bounded linear operators is given whose elements commute with the bilinear integral operator. This semi group o[ commuting operators provides a method for generating new solutions [rom known ones. §l.
INTRODUCTION
It has been shown [5] that the Tjon-Wu model for the Boltzmann equation represents an evolving, spatially homogeneous and isotropic gas interacting through binary collisions. It is a convenient model to study [or it is equivalent to other, more physical ones. And, most important to this paper, the conceptually simpler Tjon-Wu version leads to an infini te system of coupled equations [l, 2]. While the coupling is nonlinear, it is clear that the infinite system has a solution that evolves in time.
However, to maintain the equiva-
lence of the infinite system and the Tjon-Wu model, rates of growth for the infinite system are required. We announce improvements on the results of [3]. By examining the spectrum of a particular linear operator, we demonstrate the existence and provide a study of densely defined operators T which commute with the integral operator in the Tjon-Wu equation. 45
46
M.P. BilRNSLEY et al.
The nonlinear eigenvalue property to which we refer is 00
[
x
e-Y Y o y - [ dz L (y-z)L (z) Y 0 P q
1 p+q+
= - - 1 e-
x
L,
p+q
(x)
where {Ln}~=O is the sequence of Laguerre polynomials. Identify -x e Ln (x), n 20, so that
[y ( ( [ x dy y 0 dzo p y-z)0 q z) = p+q+l 0p+q(z) Also, we have [
° (s)O o p
q
s (s)e'dx -_ \
i.f p
0
if p
1
l'
q
The space L
2
is. then,
=q
the closure of the span of {On }~=O with the norm arising from the inner product indicated by the orthogonality relations [or {O }'" 0 n n= Theorem 1.1 ,00
Lp=O
ct
p °p'
(i)
If
is in L2 and {a }'" 0 is in ;>,2 such that o p p= then these are eCluivalent: (3]
U
o
(0,"') ~ L2 which is a solution for the
there is a function u:
Tjon-Wu equation with u(O) (ii)
U
=
u
o
'
and
there is a sequence {cp(t)};=o of number functions such that c : n
(0 ,00)
m
~
is a solution for
c' + c n n
(TW)
and, for each t, {c (t)} p
It is this coupled, infinite system in ;',2 which we consider.
§2.
THE EXISTENCE OF EQUILIBRIA AND NEARBY SOLUTIONS Hereafter, we consider CTW) in the Hilbert space ;',2 of square-
summable sequences.
Also,
{p};=o denotes the usual basis in ;>,2.
'" Lemma 2.1. If x and y are in;', 2 and {zn}n=O is given by zn 2 1 n+1 Lp=O xn_pYp' then Z is in;', and I z I <:; I x I I Y I .
,n
Proof. '"
n
By the Cauchy-Schwartz 2
2
In=oIp=o xn_pY p = Definition.
2
inequali ty, zn
<:;
1xl· 2 1y 12 .
The function A on ;',2 x;', 2 is given by
1 ,n 2 2 n+T Lp=O xn _p Yp
and
47
BOLTZMANN EQUATIONS IN HILBERT SPACE
A(x ,y) (n)
n+l Lp=O x n _p y p
The [unction A is bilinear from ~2x£2 into £2 and
Lemma 2.2. IA(x,y) I
,n
1
=
Ixllyl·
<;
Theorem 2.3. If Ixl < 1, then there is only one function u: [0,00)-> [2 which is a solution [or u' + u = A(u,u) with u(O) = x. Moreover, Iu (t) I
I u (0)
<;
I
exp ( [ I u ( 0) I - 1] t) .
Remark. The inequality of Lemma 2.1 allows this improvement of the weaker theorem of [3]. Otherwise, the proofs are similar. Theorem 2.4. Suppose y is In £2. These are equivalent: A(y,y), and (ii) either there is a number a such that -1 n Yn = a [or all n or Yn = 0 fo r all n.
(i ) <
Y a < 1 and
Remark. Proof of the previous theorem appears in [3]. It identifies the class of equilibrium solutions [or (TW). We see in the next theorem that solutions having initial values on a line through the origin and through an equilibrium point can be given explicitly. Indeed, their trajectories remain on that line. Definition. I[ -I < a < 1, then we identify a in £2 as the equi112 brium point {l,n,n , ••• } and identify Sa as the line containing 0 and a: S
{ c~: c
a
E
IR } .
If c is a number, then there is a Suppose -1 < a < I. solution for CTW) such that ufO) = ca. In fact, u(t) = ~/(l+bet) for some b. Furthermore, lim i f c > 1 then -1 < b < 0 and lu(t) I = (a) U-£n(-b) if 0 < c < 1 then 0 < b and limiu(t) i = 0, and (b) t+oo
Theorem 2.5.
(c)
if c
0
<
then b
<
-1 and limlu(t)1
=
O.
t+oo
Indication of Proof. I[ c is a number, b is defined by c and u(t)
~
=
t
=
l/(l+b),
~
a/(l+be ), then ufO) = ca and u' + u = A(u,u).
§ 3.
ASYMPTOTI C BEHAVIOR OF SClLUTI ClNS In this section, we obtain solutions near the equilibrium solutions and near the lines S The proof of Theorem 3.1 is found in a
[3] .
48
M.F. BARNSLEY et al.
Theorem 3.1.
There is an interval I
such that 0" I
tive function R on I such that i[ a is in I
c
(-1,1)
and a posi-
and x is in the convex
set C = {[: O> = 1, <[,1>1> = a, and If-al ,; R(a)}, then there a is only one function u: [0,00) ~ C such that u is a solution of (TW) a with ufO) = x. Moreover, if lu(O)-al < RCa), then lim uet) = a. t->-oo A
Remark.
For the physical significance of <[,1>0> = 1 and <[,1>1> = a
see, for example, Lemma 3.2. Yl = Yo
[3].
Suppose that each of x and y is in .(2, and xO=xI=O= 1 Then IA(x,y) I ,; "4 Ixlolyl·
)1 2 ,00 (1 ,n-2 x,y = Ln=4 n+l Lp=Z ,00 n+l n 2 1 Ln=O 2I.p=O(x2+n-pyp+2)'; 16 (n+ 5)
Proof.
IA(
Theorem 3.3.
Suppose b
~
0, x is in .(2 such that <x,1>O> = l/(l+b),
<x,1>I> = 0, and
IX-1>O/(l+b) I < 4(1+3b)/3(I+b).
one [unction u:
[0,00)
Then, there is only
~
.(2 such that u is a solution o[ (TW) and t Moreover, for that solution, lu(t)-¢o/(l+be ) I 4- 0 as t
ufO) = x. increases.
Indication of proof.
Let b and x be as supposed and m be a positive
number such that m < 4(1+3b)/3(I+b). set described by C
Let C be the closed, convex
{z: = 0, I> = 0, and
J(t) be the function given by J(t)z = A(z,z) Then J(t): C
~
Izl ,; m}. Let t + ZA(¢o,Z)/(I+be ).
C and, for each z in C, J(o)z is integrable on com-
pact intervals.
is a
function v:
By Theorem 1.4 of [4], if z is in C then there 2 ~.( such that v' + v = J(t)v, v(O) = z. Let t x - 1>O/(I+b) and let u(t) = vet) + ¢o/(l+be ) with v as above. t u' + u = A(u,u) and ufO) = x. Also, lu(t)-1>O/(I+be ) I ,; IX-1>O/(I+b)l o exp(ct) where c =} Ix-¢O/(I+b)1 - (l+3b)/3(l+b) <
z =
Remark. all t
>
[0,00)
O.
When b = 0 we have containment in .(2 of the solution for 4 IX-1>ol < 3"' This improves the
o with initial value x where
previous estimate Remark.
Then
IX-1>o I ,; .7085 which was obtained in [3].
In a similar manner,
it can be shown that if b and a are
related by b ~ 0 and 21~1 < 3(I+b) and if Y = a./(l+b)
then we have
this stability result:
I f Ix-yl
is only one function u:
[0,00) ->- .(2 such that u is a solution of (TW)
and ueO) = x.
Moreover,
< 4(3(1+b)-21&'1]/3(1+b)
then there
lu(t)-~/(l+bet) I 4- 0 as t increases.
49
BOL TZMANN EQlJA TIONS IN HILBER T SPACE
§4.
LINEAR OPERATORS WHICH COMMUTE WITH A In this section we demonstrate that there is a two-parameter semigroup of bounded linear operators T b such that a, ACTa , bCx),T a, bCy)) = Ta, bCA(x,y)) for all x and y in a dense set. Al so T b T d = T b bd' a,
C,
a+ c,
We use the following notation: Aa is the linear operator A(;;,.) and N(x) = min{n: x(n) fO} for x f 0 and x in ,[Z. T__h_e_o_r_e_m__4~._1.
Suppose -1 < a < 1.
Then A
is a one-to-one, Uilbert2 a Schmidt operator with IIA a II s 1/(l-a). Furthermore, the non-zero spectrum of A is {lin: n = 1,Z,···} and each eigenvalue has multiplicity 1.
-
Z
Proof. From Lemma 2.2, we have that A is a bounded, linear operator 2 2 a and IIA 11 25 1;;1 = l/(1-a). To see that A (y) f 0 i f y f 0, let a a 1 n = N(y) and note that = A(a,y) (n) = n+l Yn f O. To see ----
A
that Aa is Hilbert-Schmidt, we sum:
,00 ,P Lp=OLq=O
1 [p+l a P - q 12
S
'IT
21
a I 2 16.
rfOp=OL~=O
,¢p>2 =
Finally, we consider the spectrum.
Suppose that A is an eigenvalue, x is an eigenvector, and n = N(x). Then Ax (n) = A (x) (n) = !l In 0 an-px = ~l x(n). Thus A = l/(n+l). n p= p n+ a 1 On the other hand, if n is a nonnegative integer, x f 0 and n+l x= Aa(x) then N(x) = n. Also, for p > 0 , ~l x = n+ n+p 1 ,n+p n+p-k 1 ,p-l p-k n+p+l (x n +p + Lk=Oa x n +k )· Thus, x n +p n+p+l Lk=Oa xk n+l ,p-l p-k P L.k=Oa xn+k' And, we see that upon choosing xn ' x is completely oo P¢ determined and is x n Lp= o(p+n)a n n+p Corollary 4.2. If -1 < a < 1 and x f 0 then A(a,x) = AX if and only if there is a nonnegative integer n such that A = l/(n+l) and x oo P¢ cL p= o(p+n)a n n+p for some c f O. Remark.
In a similar manner it can be shown that if x is in ,[2 and
Xo f 0 then Ax ~ A(x,') is a one-to-one, Hilbert-Schmidt operator with I IAx\ I S Ixl. As before, the non-zero spectrum of Ax is {x0/n: n = 1,2,"'} and such members of the spectrum are eigenvalues of multiplicity 1. If Xo = 0, then Ax is quasi-nilpotent; that is, its spectral radius is zero. Theorem 4.3. Let T be a linear operator on D, the span of {¢ p }oop= 0 ' such that A(Tx,Ty) = TA(x,y), for each x and y in D. Then, there
50
M.F. BARNSLEY et al.
are numbers a and S such that
lal
< 1 and T(x) (n)
,n (n)an-PSpx n = 0,1,2, ••• Lp=O P p , Proof.
1fT has the commuting property and is 1 inear, then using the
nonlinear eigenvalue l)rojierty, A(H
H ) = __ 1_ T(w ). In parm' n m+n+l m+n ticular, A(TwO,TwOJ = T(w ) so that either T(¢O) = 0 or, by Theorem O 2.4, T(w ) = & for some a in (-1,1). Furthermore, A(T¢n,T¢O) O ~1 T(¢ ) so that j f T(w O) = 0 then T(¢ ) = 0 for all nand T" D. n+ n . n 1 If T t 0 and T(w ) = &, then A(H ,&) = --1 T(w ) so that, by Coroln n+ n O
snLp=O ,00 (p+n)aPw n p+n
lary 4 2 T(w) = ., n
°
= B ,'" (P)aP-nw for some senLp=n n p
quence {s }"" and lal < 1. To determine {S }oo_O' we examine n n= n n,HI) (n+l) = ~7 T(q, l)(n+I) = A(H ,H ) (n+l) = n n+~ n+ n l _1_ ,n+l Tew ) '1'(q,) But n+2 Lk=O n n+l-k 1 k .
A(H
0 if k = () k-n
T(W ) (k) 1 and
B1(~)a
Ia I
Ibl
a)
If
If lal + Ibl
+
< 1 then T
k
n +1
,;
(n+l-k)a l-k if k = 0 or 1. n
Suppose a and b are numbers and
b)
,;
0 if k > 1
T(Wn)n+l-k
Bn
Theorem 4.4.
if 1
a,b
lal
< 1.
is Hilbert-Schmidt.
~ 1 then Ta,b 1S bounded and
IITa , b 1l2 ,; l/[l-laIClal+lbl)] c)
If lal
+ Ibl
> 1 then there is x in (2 such that x is not in the
domain of T a, b Proof.
(a)
To see that T
a,
,00 ,n (( n ) an-PbP) 2 Ln=OLp=D p
b is Hilbert-Schmidt, sum: ,;
C,n Cn) lal n - P lbI P )2 L.n=O Lp=O p
\,00
00 2n Ln=o(lal+lbl) <
provided lal
+ Ibl
< 1.
00
51
BOLTZMANN HQUATlONS IN HILBERT SPACE
To sec tha t T , b is a bounded opera tor, aga i 11 sum: a
(b)
,00
,n 11 n-p p 2 Ln=O(Lp=UCp)a b Xp )
'" rn
n= 0
On jl =
(11)[a[n-p.[h[p).On (n)[a[n-p.[b[px2) JF () jl P
0 P
I'Dn=O ([a [ + [ b [ ) n Lnp=O (n) [ a [ n - p [ b [ 1\ Z p p t"-or"- ([a[+[b[)n(n) [a[n-p[b[pxZ pn-p p p
L;=o([a[+[b[)p[b[Px~.L~=o
en;p) ([a[([a[+[h[))n P
2 1. 00 [b (3+b')J l-[a[([aj+jb[) Lp=O 1- a (a + hT xp 2 '" Ixl provided [et[ < I and [a[ l-Ia C al+lbl)
+ [hi
Ib\C\a\+[b[) [I (That l-[a[(la[+[bl) '" ] is l'l{uivaJC'llt to a Finally, to see that if \al
(c) T
+ [bl
+
'" 1. \ [ h '" 1.)
> 1 and lal
< I then
b is only densely defined, let a be such that -1 < a <
a, \a+ba[
~
A
Then note that T(a) = (a+boH C.
J.
Theorem 4.5.
The collection {I'
a,
b:
[a[ + [b[ '" 1,
parameter semigroup of bounded linear operators.
Ta,b l' c,d Proof.
=
and
A?
lal f l} is a twoSpecifically,
Ta+bc,bd'
Since T
a,
bT
x and y such that T
c, G,
d commutes with the bilinear operator there arc bT,., = T c,u
X,Y
Also, a+bc=
3,
bT'd(¢Ol'¢l>= C,
x, y . = x and bd = 1> = 1> = y. Remark.
Many of these same results hold for A(x,y) (n) = I'n a = 1 Th'l5. provl. d es 3 s t u d Y Lp=O"np .
I'n a np x n-pyp were h lp=O a np >- 0 all,1•
of c' + c = In oa c c c (0) = a i.n .£.2. Or, as in Theoreml.l, n n p= np n-p p' n ' n dU y 2 8t (t,x) + uet,x) = ! dy! dz k(z,y)u(t,y-z)u(t,z) with u(O,x) in L , x 0 for appropriate choices of k.
52
M.P. BARNSLEY et al.
REFERENCES 1.
M. F. Barnsley and H. Cornille, General Solution of a Boltzmann Equation and the Formation of Maxwellian Tails, Proc. Royal Soc. London A, 374 (1981), 371-400.
2.
M. F. Barnsley and H. Cornille, On a Class of Solutions of the Krook-Tjon-Wu Model of the Boltzmann Equation, J. Math. Phys. 21 (1980), 1176-1193.
3.
M. F. Barns1ey, J. V. Herod, V. V. Jory, and G. B. Passty, The Tjon-Wu Equation in Banach Space Settings, Journal of Functional Analysis (To appear).
4.
H. Brezis, OperateursMaximaux Monotones, North Holland Publishing Company, Amsterdam, 1973.
S.
J. A. Tjon and T. T. Wu, Numerical Aspects of the Approach to a Maxwellian Distribution, Phys. Rev. A 19 (1979), 883-888.
Spectral Theory of Differential Operators I.w. Knowles and R. T. Lewis (eds.) © North-Holland Publishing Company, 1981
SOME PARTIAL DIFFERENTIAL OPERATORS WITH DISCRETE SPECTRA
v.
John
Baxley
Department of Mathematics Wake Forest University Winston-Salem, North Carolina U.S.A.
We study selfadjoint realizations of the formal differential -1
[(Plu) + (P2u ) 1 in the weighted Hilbert x x Y y space Lm (S'l) where rl is the square domain (0,1) x (0,1) . Assuming
operator Tu 2
=
-m
m, PI' P2 are positive and reasonably smooth and that singularities of T occur only along the boundaries x
=
0 or y
=
0, a variety of
strictly positiveselfadjoint realizations of T are constructed, each of which, with a further integrability condition on the coefficients, has a discrete spectrum.
1.
Let T be the formal differential operator
(1)
1
TU
If rl is a domain
m
+ (P2 uy ) Y 1.
[(Plu)
x x
in~2, this formal operator may give rise to a variety of self-
adjoint operators in the weighted Hilbert space L2(rl) consisting of all m
measurable complex-valued functions u defined on rl for which Ilul ~
=
(II
I ul
2
1/2
)
m dxdy
rl
<
The inner product is of course given by (u,v) =
II
u
v m dxdy,
2 for u,v ELm (rl).
rl We are interested in situations for which rl is singular and we wish to construct operators with discrete spectra. actually have compact inverses.
The operators we consider will
Since our goal here is to exhibit a tech-
nique for the construction and investigation of such operators, we shall treat the simple domain rl
=
(0,1)
x
(0,1).
restricted to the boundaries x
=
We shall assume that singular points are
0 or y
=
0 and our basic assumption will be
that (2)
1
dt) dxdy <
PI (s,y)
53
54
JOHN BAXLEY
The well-known contrast with the one-dimensional case should be kept in mind.
In that case, every selfadjoint extension of the minimal operator has
the same continuous spectrum and, hence, if one selfadjoint extension has a discrete spectrum, all selfadjoint extensions have discrete spectra. dimensions, this result is no longer true.
In higher
Thus it becomes necessary (and
poses a fascinating problem) to examine separately every selfadjoint realization of
L.
Criteria for discrete spectra have long been of interest; the two papers of Friedrichs [4],
[5] describe the status of the problem at mid-twentieth
century. 2.
It will be clear to anyone who is familiar with Baxley [1], [2] or Rollins
[8] that the
pr~sent
sional experience.
construction and methods are motivated by that one-dimenThe criteria presented by Rollins were close in spirit, though
not method, to the criteria given by Eastham [3].
More recent one-dimensional
criteria have been given by Hinton and Lewis, e.g.
[6].
The reader should also
compare the recent work of Lewis [7] on partial differential operators, which was discussed at this conference. Because the criteria presented here are the natural two-dimensional analogues of the one-dimensional criteria in [1], [2],
[8] and the strategy used here is
the same, although technically more complicated, we state for comparison those criteria. Let Tu; - ;(pUI)I, 0 < x < 1.
Let
C~(O,l)
be the class of infinitely
differentiable functions with compact support in (0,1).
Assume that m, p' are
continuous and strictly positive on (0,1] and that (3) L may be singular at x ; O. Put Lu; Tu for u E C~(O,l). Then L is a 2 symmetric, positive operator in the weighted Hilbert space L (0,1) and the m Friedrichs extension of L has a compact inverse and hence a discrete spectrum.
Thus
The
condition (3) should be compared to (2).
In [2], a second selfadjoint
extension of L is also considered and furnishes motivation for the various operators considered below. 3.
Let f be the boundary of the rectangle r.! ;
points of f with either x ; 1 or y ;
1.
(0,1) )( (0,1).
Let f2 ; f - fl'
Let fIe f be the
Singularities will
be confined to f
. 2 We shall assume that m E C(st U f 1)' Pl' P2 E C I (st u f 1) and that m, Pl'
P2 are all strictly positive on st U fl'
Thus, we allow any or all of the points
of f2 to be singular for the formal operator m, Pl' P2 may tend to 0, regular case.
00,
or oscillate.
L
given in (1), for at these points
Note that our analysis includes the
55
PARTIAL DIFFHRENl'L1L Ol'l,RA TORS WITH DISCRl!Ti": SPHC]X1
Our plan is to describe a variety of selfadjoint realizations of T, which will be obtained as Friedrichs extensions of restrictions of T to different initial domains.
{x: u
For this purpose, choose any subset fO of f 2.
{y: (O,y) E f O}.
(x,O) E f O} and BO =
He define DO C
L~(!:t)
Let AO = as follows:
E DO if and only if (a)
u E c'-"(!:t u f
(b)
u = 0 on fl
(c)
l
there exists
)
°
0
(depending on u) such that
if 0 < x < 0, x < y < 1, then 0 for y f/; BO' or ux(x,y) = 0 for y E BO' u(x,y) < x < 1, then Y 0, u (x,y) = 0 for x E AO' u(x,y) = 0 for x f/; AO·
1£0 < Y <
°
Y
For 0 < 0 < 1, let !:to = (0,1) x (0,1) C!:t.
If uEDO and 0 < 0 < 0
0
(see (c», i t
follows that either u or the normal derivative of u is zero at each boundary point of !:to. Let Lu = Tu for u E DO where T is given by (1). Then L is densely defined 2 in L (!:t). He shall see below that L has a Friedrichs extension F. Since in m general the domain of L varies with f ' it is expected that many different O Friedrichs extensions will be obtained. However, it is possible that different L's have the same Friedrichs extension.
In extreme cases, it may happen that
the Friedrichs extension is the same for every L.
The discussion on pp. 248-249
of [2] is of interest here. In the case that fa is empty and T is actually regular, F is the classical Dirichlet operator with zero boundary conditions. problem.
Otherwise, we get a mixed
If fO = f2' and T is regular, F is the operator with these boundary
conditions:
u
=0
if x
=1
or y
= 1, ~~
=
0 if x = 0,
~~ =
o<
y
~
1, then
r
PI (s,y)
lu (s,y)1 s
P2(x,t)
lu (x,t) 12 dt t
0 if Y = O.
He proceed with the proofs.
4.
Lemma l.
For u E DO' and 0 < x < 1, (1)
iu(x,y) 12 <
rr x
(ii)
lu(x,y) I 2
':
Y
Proof.
He prove only (i).
1
PI (s,y)
1
P2(x,t)
ds
0
(1
dt
J0
2
ds
Using the fundamental theorem of calculus and
Schwarz's inequality, we have
II
lu(x,y)1 2 = 1
us(s,y) dsl 2 <
x
II
and (i) is immediate. Lemma 2.
1
( x PI s,y)
L is symmetric and positive.
ds
II
x
Pl(S'y)
1us(s,y) 12 ds
56
JOHN BAXLEY
If u,v E DO' we may choose 0 > 0 small and integrate by parts twice on
Proof. ~o
o~O,
using Green's theorem, after which letting
thus L is symmetric.
we get (Lu,v)
=
(u,Lv) and
In the same way, but only integrating by parts once, we get
(Lu,u)
(4)
for u E DO' ufO. It follows from Lemma 2 that L has a Friedrichs extension F. have not used the condition (2).
Thus far we
However, condition (2) is crucial for every-
thing that follows. If Q* C Q is measurable, put
Lemma 3.
M~ If u E
«(i*)
=
fJ* m2 (x,y)U: PI (!,y)
d, ( (
P2(~'t) d~
dxdy.
DO' then
If
lul
2
m dxdy
~ M«(i*)
(Lu,u).
(i* In particular, (u,u) Proof.
~
M«(i) (Lu,u), for u E DO'
From Lemma 1, we have
lu(x,y)1
2
~
1 (Ix1 Pl(s,y)
(f
ds fl 1 dt) 1/2 y P2(x,t)
l
o Pl(s,y)
lus(s,y)1
2
x
II
ds
0 P 2 (x,t) lut(x,t)1
2
dt
) 1/2
Thus by Schwarz's inequality and (4) rJ*lu l2 m dXdYY
~M2(Q*)
~M2(~*) fJ~>1(S'Y)IUs(S'Y)12
If
Pl(s,y) lu (s,y)1 s
2
Q
dsdy
If
ds f>2(X,t)lu t (x,t) 12 dt) dxdy
P2 (x,t) lu (s,t)1 t
2
dxdt
~
~ M2«(i*) (Lu,u)2, and the result follows. Lemma 4.
For each u in the domain of F, (u,u) ~ M(m (Fu,u), lIull ~ M(.Il) IlFull.
Proof.
For such u, there exist un
E
DO such that Ilu
n
- ull ~ 0 and (Lu , u )+(Fu, u) n
n
By Lemma 3, (un'u ) ~ M(Q) (Lun,u ) and taking limits yields the first n n desired inequality. The second follows from Schwarz's inequality.
as
n~.
Corollary.
If A is a point in the spectrum of F, then A ~ (M(Q))-l.
F has a bounded inverse.
In particular,
57
PARTIAL DIFFERENTIAL OPERATORS WITH DISCRETE SPECTRA
Note that this estimate for the lower bound of the spectrum of F is rather crude in that it applies to every selfadjoint operator F obtained by our procedure. Theorem.
The partial differential operator F has a compact inverse and hence a
discrete spectrum. Suppose un is in the domain of F and IIFu II = 1 for each n = 1, 2, ••.. We
Proof.
n
L2(~), and the completem
shall show that {u } has a subsequence which is Cauchy in ness of
2
Lm(~)
n
gives the desired conclusion.
We may choose un E DO for which II u for n
=
1,2, ....
n
- v
n
II
<
I (Lvn,vn )
1, n
- (Fu ,u ) nn
<
1n
It follows from Lemma 4 and the Schwarz inequality that I (Lv ,v ) I < M(~) + 1, for n
(5)
I
n
n
-
=
1, 2, ....
Put Vn(x,y) =
II
vn(x,t)
d~ for
(x,y) E
~
u rl.
y
Using a rather trivial form of Green's theorem, Schwarz's inequality and (4), we have
1 dxdy Pl(x,y)
dxdy
1 dxdy (Lv ,v ) PI (x,y) n n
and
1 dxdy (Lv ,v). Pl (x,y) n n Therefore
[Il I
X
- V (xl'Yl)I 2 .::. n
Yl
2
xl
( 1 ) dxdy PI X,y
+
I Ill] Y 2
Yl
x2
P
l
( ) dxdy X,y
.
(Lv ,v). n
n
It follows that {V } is uniformly bounded and equicontinuous on compact subsets of n
58
JOHN BAXLEY
Q U fl'
Using Ascoli's theorem and a diagonalization argument to pass to a sub-
sequence if necessary, we may assume that {V } converges uniformly on each compact n
subset of Q u fl'
°
Now let E.> be given and let Q = (0,1) x (0,1). o 2 2(lv 12 + Iv 1 ), then using Lemma 3 and (5), n m
Since Iv
II
- vm12 m dxdy -< 2 Ivn 12 m dxdy + 2 Q-Qo ~2M(Q
n
- v 12 < m
-
m dxdy
-Qo) [(LVn,V ) + (Lvm,v )] n m
Because of (2), we may choose 0 E (0,1) sufficiently small so that ?
- v 1- m dxdy < E/2, for all n.
(6)
m
Let W (x,y) n
Vn(x,y) - Vn(X,O).
Then
ClW ClV -E. = --..!lc = -v Cly Cly n and {W } converges uniformly on Q ' o n integrating by parts,
II Q
Iv
o
K
II
n
max{m(x,y): (x,y)
Letting K
- v 12 m dxdy < -K m -
II(foy
(W
n
- W )) m
(vn - vm)
e
Qo} and
dxdy
Q dV dV D ( ---E. - --.!I' ) dxdy.
(wn - wm) dY
Cly
Q
o
Thus, using Schwarz's inequality, (4), and (5), dV I d/
dxdy
(7)
JJ Q
o
Combining (6) and (7), we have
IVn - v m I
2
m dxdy < E/2.
dV 2 - dyml dxdy
PARTIAL DIFFERENTIAL OPHRATORS WITH DISCRETE SPECTRA
59
v 12 m dxdy < E for n,m > N. m
-
Thus {v } is a Cauchy sequence in L2(1t) and since Ilu - v 11< l, for each n, n m 2 n n n then {u } is also a Cauchy sequence in Lm(It). n
References [1]
Baxley, J. V., The Friedrichs extension of certain singular differential operators, Duke Math. J. 35 (1968) 455-462.
[2]
Baxley, J. V., Eigenvalues of singular differential operators by finite difference methods, I, II, J. Math. Anal. Appl. 37(1972)244-254, 257-275.
[3]
Eastham, M. S. P., The last limit point of the spectrum associated with singular differential operators, Proc. Camb. Phil. Soc. 67 (1970) 277-281.
[4]
Friedrichs, K. 0., Criteria for the discrete character of the spectra of ordinary differential operators, in: Courant Anniversary Volume (Interscience, New York, 1948).
[5]
Friedrichs, K. 0., Criteria for discrete spectra, Corom. Pure Appl. Math. 3 (1950) 439-449.
[6]
Hinton, D. B. and Lewis, R. T., Singular differential operators with spectra discrete and bounded below, Proc. Royal Soc. Edinburgh Sect. A 84 (1979) 117-134.
[7]
Lewis, R. T., Singular elliptic operators of second order with purely discrete spectra, preprint.
[8]
Rollins, L. W., Criteria for discrete spectrum of singular selfadjoint operators, Proc. Amer. Math. Soc. 34 (1972) 195-200.
This page intentionally left blank
This Page Intentionally Left Blank
Spectral Theory of Differential Operators I.W. Knowles and R. T. Lewis (eds.) © North·Holland Publishing Company, 1981
SPECTRAL THEORY FOR HERMITEAN DIFFERENTIAL SYSTEMS Christer Bennewitz Department of Mathematics University of Uppsala Uppsala, Sweden The notion of hermitean differential systems is due to Schafke and Schneider who discuss two kinds of such systems; right-definite and left-definite. The spectral theory in the right-definite case is essentially complete. The left-definite singular case was treated only with certain restrictions in general excluding systems obtained from pairs of scalar operators. This paper describes a spectral theory which does not suffer these restrictions. a.INTRODUCTION In a number of papers, starting with [6J, Schafke and Schneider and later Niessen have studied spectral properties of so called S-hermitean differential systems. These systems are accessible to a spectral theory if they are either right- or left-definite (see section 1). The first paper dealt, mainly by algebraic means, with the regular case, giving discrete spectra. The singular case for right-definite systems was treated e.g. in [4] and for left-definite systems in [7]. The results in the right-definite case are essentially complete whereas the left-definite case has been successfully handled only under certain restrictions. The systems obtained from pairs of scalar differential operators, the spectral theory of which was given e.g. in [5] and [2], do not in general satisfy these restrictions. See the discussion in [3]. This paper gives a unified treatment of right- and left-definite systems which is sufficiently general to include the systems derived from scalar equations. The method is essentially that of [1] and seems simpler than the techniques of Schneider and Niessen in [4] ,[7J. Only the characterization of possible selfadjoint operators derived from a differential system is presented here. The important problem of giving appropriate integral transforms and expansion theorems will be treated elsewhere. 61
CHRISTER BENNEWITZ
62
Section 1 gives basic definitions and hypotheses. In section 2 is derived the characterization of the appropriate selfadjoint operators from the basic theorem, the proof of which is given in section 3. 1 . BASIC DEFINITIONS Let p,q,s and t be complex MxM matrix-valued functions defined on a real interval I. For simplicity it is assumed that p is continuously differentiable and q,s,t continuous in I although these requirements can easily be slightly relaxed. Furthermore it is assumed that throughout I holds 1.
p*= -p, q*= q and t*= t
2.
P is invertible and tq nilpotent
3.
p' = ps + s *p .
The differential operators S, Q and T are then defined as follows: Su
CI
u'+ su
Qu = qSu - pu
Tu
CI
tQu
Here, and in what follows, u (and later v, w etc.) denotes a complex
Mx1 matrix-valued continuously differentiable function in I. With these definitions the differential relation Su=Tv is called Q-hermitean in accordance with the terminology of [6] ,[4], [71. Note that the conditions 1-3 imply that pS is formally symmetric and that in the operator
S-AT, AE [, the coefficient of the derivative is invertible
everywhere in I so that standard existence theorems apply for the equation (S-\T)u = v. For a compact subinterval J of lone has (1 • 1 )
J(Qv)*Tu
JCTv)*Qu
J
J
/CQv)*Su = JCSv)*Qu + [v *pU]J . J J
(1 • 2 )
The first formula is obvious. The second follows on integrating by parts and [ .. ']J denotes the out integrated part. Denote the integral in C1.1) by Cu,v)T , J
.
~§fi~i!i2~~ The relation Su=Tv is called Eigb!:9§fi~1!~ if 1
1.
(u,u)T,J > 0 for UE C (1) and every Jcc 1.
2.
(u,u)T,J > 0 for non-trivial solutions of Su=o and sufficiently large J Cc I.
It is easily seen that this implies that
t~O
and that in the defini-
tion we may equivalently replace Su=D by SU=ATu for some, or all, AE a;
63
SPECTRAL THEORY FOR HERMITEAN DIFFERENTIAL SYSTEMS
It is considerably more involved to make a sufficiently general definition of left-definiteness. To begin with we assume that M=2m is even and that there is a continuously differentiable function k such that p a k-k*
and k has rank m throughout I. Assuming this put
(u,v)s J a j{(Qv)*Su-(v*ku)'} , J
1
u and v in C (I)
It is then an immediate consequence of (1.2) that ("')S,J is hermitean. Let
cl
1
be those u in C eI) for which Su is in the span of 1m t
and Ker k* everywhere in I. Note that u is certainly in solves Su=Tv for some v in C1 (I).
cl
if it
1.
. C*1 and J ( ) S,J ,;: Of or every u In u,u
2.
(u,u)s,J > 0 for non-trivial solutions of SU=ATu for some AE
[C
and every sufficiently large J
CC
Cc
I
,
I.
1 The reason for not simply requiring positivity on C (1) is that this would certainly exclude most interesting examples, notably the canonical systems considered in [7] and those derived from scalar equations. Suppose namely that Sy = I(-1)j(p./j»(j)
o
]
where all Pj';:O, Pm>O and PDf 0, and that
T is
formally symmetric of
lower order. Then the equation Sy=Tz may be equivalently written (see [31 section 2) u' + [- c *
(1 • 3 )
A
HJ u C
=
[0 0Jv'
+
G* 0
HG OJ [ B+CG 0 v
where A,B,C,G,H are mXm matrix-valued and A,;:O, 1m G C 1m H. Put s a [-:* where
q
~)
, t
a
[
G~H
HG B+CG+G*C*
J
, k
H~O,
B*=B and
[~ ~J
, q =
[~ ~J
is chosen so that qH is the orthogonal projection on 1m H.
Then (1.3) is the corresponding Q-hermitean relation, u and v are in
cl
and the system is left-definite according to our definition (with j 2 Jl:P.I/ )1 ), but ("')S,J is not positive ]
(u,u)S J coinciding with on all' of C1 (1) • B~~~r~~
The spectral theory in [7J is carried out for systems of the form (1.3) with G=O and B satisfying -pA,:SB,:SpA for some p ElL
64
CHRISTER BENNEWITZ
2,SPECTRAL THEORY Let ( " ' ) J denote ("')T,J in the right-definite and (·,·)S J in the 1 be C (I) in the right-definit~ and as left-definite case and let
cl
in section 1 in the left-definite case. Considering the part of C*1 giving a finite value to (u,u)I and introducing the quotient with respect to elements with vanishing norms we obtain after completion a Hilbert space H with norm IUl 1
r
1
Eloc = {(u,U)E C*xC* EI
1:1
{(u,u) E E
D\
1:1
{(u,\u) EEl}
E\
D\
+ D\
=
~r
Su=:Tu}
loc
for 1m \
t 0
BJ(U,V) =-i(u,v)J-(u,v)J) EO
I
Br(U,E ) = O} I One might view EI as the maximal and EO the minimal relation associated with S and T in the norm I . I I The basis for the spectral 1:1
{UEE
for U=(u,u), V=(v,v)
I
theory is then given by
!l:!§:9!:§JE':' 1 • Er = EO .j. EA
as a direct sum
1
2. For VE C* with Ivlr
fO there is an element
(u,\u+v) in Er Concretely 2. means that there is a solution u of SU-ATu=Tv with finite norm. The proof of the theorem is given in the next section. In the following discussion we will use the theory of symmetric relations on a Hilbert space. A presentation adapted to our present needs may be found in section
of [2J.
Let E be the closure of EO in H modulo elements with zero norms. Then
°
it follows from BrCEO,E ) = that E is a closed symmetric relation r on H and the closure Er of Er modulo elements with zero norms is contained in E*. rn fact, this closure equals E* which follows from
( 2•1 )
E*
because clearly
span{E,E\} C £1 C E* . To prove (2.1), note that
1:1
E .j. E\ ' 1m A fO (a topological direct sum)
from the theorem follows that for 1m \
f 0 and v E
cl
with finite norm
there exists an element (u,\u+v) in EO precisely if (v,w)r=O for all (w,\w) in D\ • This is because determining U=(u,\u+v) in Er by the theorem we have
BrCU,(w,\w»
1:1
-i(v,w)r and by adding an appropriate
element of D\ to U we may achieve Br(U,D\)=O. For such a U in EO one has
65
SPECTRAL THEORY FOR HERMITEAN DIFFERENTIAL SYSTEMS
o
= Br(U) = 21m \(u,u)I + 2Im(v,u)r
so Cauchy-Schwarz' inequality implies IUII~IIm\I-1 Ivlr . Taking closure it thus follows that for each VE H there exists (u,\u+v) in E provided only (v,w)r=O for each (w,~w) in D~. Since D~, being finitedimensional, is closed it follows ([2J
lemma 1.3) that
D~
is the de-
ficiency space of E at ~. Now \ is arbitrary non-real so (2.1) follows ([2J theorem 1 .4). Having thus identified the deficiency spaces of E the abstract theory of symmetric relations completely characterizes the maximal symmetric extensions of E, all or none of which will be selfadjoint depending on whether dimD\=dimDr for nonreal \ or not. Each such extension is an extension of EO and has a core which is a restriction of Er (modulo elements of vanishing norm). Thus, the abstract theory gives a characterization of maximal symmetric realizations of Su=Tv in H. Furthermore, the spectral theorem for relations ([2J theorem 1.15) gives a resolution of the identity for each selfadjoint realization of Su=Tv in H.
3.PROOF OF THE THEOREM Let d\= {(U,\U)E E loc } . If U=(u,\u) then BJ(U) = 2Im\(u,u)J' Consequently B is positive (negative) definite on d\ for Im\>O «0) if J J eel is sufficiently large. In fact, d\ is m2~~m21 with this property as a subspace of E ' This follows from the Lagrange formulas loc (31) • B ( U, V) 1:1 - l' [V * pu] J J
, -l
(3.2)
[-v * ku-v * k *-] u J
where U=(u,u) and V=(v,v) are in EI
. Here (3.1) applies in the oc right-definite and (3.2) in the left-definite case. In either case it is clear that the rank of B is at most 2M (recall that in (3.2) M=2m J and rank k=m) as a hermitean form on E ' Since this is attained loc on d\+dr for non-real \ the maximality of d\ follows. Now let \ be fixed non-real and c=2Im\. For U in E
there is then loc for Jee I sufficiently large a unique UJ in d\ so that BJ(U-UJ,d\)=O since cB J is a norm-square on d\. The maximality of d\ implies that cBJ(U-UJ)~O
since otherwise cB J would be positive definite on the
span of U-UJ and d\.
66
1§~~2~
CHRISTER BENNEWITZ
Suppose U=(U,AU+V) is in E
loc
A.
UJ-+UIE d A
B.
U-UI is in EI
C.
CBI(U-U ) I
and IvII<
00.
Then
as J-+I
~
0
If U is also in EI then UI is in DA because of B .. It follows that DA is maximal positive definite with respect to cB since for U in EI I span(U,D ) = span(U-UI,D ) and hence, according to C., no extension A A of D" is positive with respect to cB I . Thus EA = D" +·D~ is maximal non-degenerate with respect to BI as a subspace of EI which is the first statement of the theorem. Furthermore, given v in
cl
the exi-
stence theorem for the operator S-AT guarantees the existence of a solution of SU-ATu=Tv, i.e. of (U,AU+V) in E
loc
so that the second
statement of the theorem follows from B. ~E92f_2f_1§~~2~ If W=(W,AW)
is in d A one obtains 2
BJ(U-W) a clu-wlJ + 2Im(v,u-w)J so Cauchy-Schwarz' inequality implies (3.3)
cBJ(U-W)+IVI~ ~
which shows that
cBJ(U-W)+lvl~ increases with J. Using BKCU-UK,dA)=O
(lei
lu- w I J -l v IJ )2 ~
0
a simple calculation shows that BK(U-W)=BK(U-UK)+BK(UK-W), It then
r
follows that if
L eKe J ee
are intervals then
(3.4)
CBL(UK-U J ) < CBKCUK-U J ) = CBKCU-U J ) - cBKCU-UK) ~ 2 2 < cBJ(U-UJ)+lvI J - (eBKCU-UK)+lvI K) .
CBJ(U-UJ)+IVI~, which is bounded from above by IVli. increases with J so that lim cBJ(U-U J ) = B ~ 0 exists. Now,
It first follows that
J+I Cfor sufficiently large Lee I) is a norm-square on the L finite-dimensional space d , A. follows. A
since cB
Letting J+I in (3.4) one obtains (3.5)
From this and (3.3) follows Iu-urlr
<
letting K+I in (3.5) one obtains C ..
00
so that B. follows. Finally,
67
SPECTRAL THEORY FOR HERMITEAN DIFFERENTIAL SYSTEMS
REFERENCES [1J
Bennewitz, C. and Pleijel, A., Selfadjoint extension of ordinary differential operators, Proc.Coll. on Math.Analysis, Jyvaskyla, Finland 1970, Lecture notes in Mathematics 419, Springer (1974).
[2J
Bennewitz, C., Spectral theory for pairs of differential operators, Ark. Mat. 15 (1977) 33-61.
[3J
Bennewitz, C., A generalisation of Niessen's limit-circle criterion, Proc. Roy. Soc. Edinburgh 78A (1977) 81-90.
[4] Niessen, H.D., Singulare S-hermitesche Rand-Eigenwertprobleme, Manuscripta math.
3 (1970) 35-68.
[5J Pleijel, A., A positive symmetric ordinary differential operator combined with one of lower order, Conf. in Spectral Theory and Asymptotics of Differential Equations, Scheveningen, The Netherlands, September 1973, North-Holland Mathematical Studies, Amsterdam (1974) [6J Schafke, F.W. and Schneider, A., S-hermitesche Rand-Eigenwertprobleme. I, Math. Ann. 162 (1965) 9-26. [7J Schneider, A. and Niessen, H.D., Linksdefinite singulare kanonische Eigenwertprobleme I, J. Reine Angew. Math. 281
(1976) 13-52.
This page intentionally left blank
This Page Intentionally Left Blank
Spectral Theory of Differential Operators I. W. Knowles and R. T. Lewis reds.) © North·Holland Publishing Company, 1981
WIRTINGER INEQUALITIES, DIP.ICHLET FUNCTIONAL INEQUALITIES, AND THE SPECTRAL THEORY OF LINEAR OPERATORS AND RELATIONS Richard C. Brown Department of Mathematics University of Alabama University, Alabama, 35486 U.S.A.
We show that several integro-differential inequalities in~luding Wirtinger's inequality, and quadratic integral inequalities involving Dirichlet functionals can be demonstrated by arguments based on the familiar fact that the norm of a bounded operator T defined in Hilbert space is the square root of the spectral radius of T* T. 1. INTRODUCTION To motivate this paper we first discuss a classic inequality often ascribed to Rayleigh and Ritz. Let [a,b] be a compact interval of the reals and let Wn ,2(a,b) stand for the complex valued functions y defined on [a,b] such that y(n-l) is absolutely continuous (AC) and yen) E L2 (a,b). Then the following result is true. Theorem 1. C1.1)
.!!
1 2
YEW' (O,ll) and yCO)
/ lyl 2
°
<
=
/Iy'
-°
y(ll)
0, then
12
with equality if and only if y = csin t. Further the "best constant" 1 in (1.1) is tEe flrst eigenvalue of the self-adjOIllt ~) 2 boundary value problem -y" = AY, yeO) = y(ll) = on W ,2(0,7I).
°
Many proofs of Theorem 1 (and closely related inequalities) are known, based on Fourier analysis, special integral transformations, the calculus of variations, or other techniques. See, for example, Hardy, Littlewood, and POlya [21], Mitrinovic' [26] or Beckenbach and Bellman [6]. Perhaps the most general reasoning however depends on some notions from the theory of operators in Hilbert space. One begins by defining L: y t-----+ y' on Wl ,2(0,1l) considered as a dense subspace of L2 (0,7I) and noting that L is a closed 1-1 densely defined operator with closed range in L2 (0,1l). Also, L has a compact (Hilbert Schmidt) partial inverse L- l generated by a Green's function and R(L) has codimension 1 in L2 (0,1l). Let P be orthogonal projection on R(L). Then the best constant C for the inequality II y II < C II Ly II (II, II :'" L2 norm) is II L- 111 = II L- 1 P II = II L- 1 P (L - 1 P )* 111/2 69
70
RICHARD C. BROWN
Since the last operator is compact, positive, and self-adjoint C is the square root of its largest positive eigenvalue X. By a theorem of von Neumann (Kato [23] ,p.27S) L*L is a s.a. operator . b y y 1-'-. -yon " { YEW' 2 2 (O,rr): yeO) = yen) = O} haVlng . glVen L-1p(L-1p)*as its inverse. By the Spectral Mapping Theorem \= l-l where l is the first eigenvalue of L*L. Further, an extremal of (l.l}(i.e., a function in D(L) such that equality holds} must be an eigenfunction of L*L. The proof is completed by observing that ! = 1 and corresponds to the unique eigenfunction sin t. Similar reasoning in fact will prove a more general result. First, however,we introduce some further notation. If H is any Hilbert space, 11·11 and [',0] will denote its norm and inner product. (We trust to the context to distinguish between various norms and inner products.) If T is an operator defined on H, o(T}, peT}, 0p(T}, D(T},and R(T} will mean its spectrum, resolvent, point spectrum, domain, and range respectively. We can now state Proposition 1. Let H, H' be Hilbert spaces and L: H + H' ~ 1-1 densely defined closed operator with closed range. (i. e., "normally solvable"). Then L*L is ~ ~.~. normally solvable operator and IIYII~)1-l/21ILYII where flO: = inf o(L*L}. li.)10 E 0pCL*L} equality is attaingd Ql ~ if and only if ~ is an eigenfunction of L*L. If )10 t 0p(L*L} then equality holds if and only if y = 0; but there exists ~ sequence ¢n £ D(L*L} with II ¢n II = 1 such that lim IIL¢n Il2 - )1(/ = 0 as n + co
A detailed proof of this result may be found in [13]. Obviously Theorem 1 is now just a corollary to Proposition 1. The same would be true for an extension of the Rayleigh-Ritz inequality due to Fan, Taussky,and Todd [19]. Theorem 2.
Ii
y E W2 ,2(O,n)
and yeO) = y(n}
;r' lyl 2 o
<
O,then
In IY"12 0
with equality if and only if y = csin t. Proposition 1 by itself however seems insufficient to prove l Theorem 3. (Wirtinger's Inequality) li. y £ W ,2(0,21r) and satisfies the condltions (1. 2)
2n J y
o
0,
yeO)
y (2n) ,
WIRTINGER AND DIRICHLET INEQUALITIES AND SPECTRAL THEOR Y
2n
2n
flY 12
o
< f
0
71
Iy' 12
The problem here is that L, if it is defined by y f---->. y' subject to (1. 2), is not densely defined so that L * does not
the conditions
exist as an operator. 0(L *L).
Thus,no meaning is assigned to L *L or to
The same difficulty would hold also for
Theorem 4.
(Fan, Taussky, and Todd [19J) If y
E:
2 W ,2(O,n),
n
y' (0) = y' (n) and f y = 0, then -0
with equality if and only if y
Theorem 5. y
E:
cos t.
(cf. Everitt [17]) Let p be
w1 ,ZCO,2n) 2n f
o
~
given positive integer,
and satisfy the conditions 2n ycosnt=f
yeO)
ysinnt=O,n
O, ... , p - l ,
0 y(2n).
Then
with equality if and only if y A different problem in the application of Proposition I is apparent for the following inequality relating the minimum of a Dirichlet functional to the infimum of the spectrum of an associated s.a. differential operator. Theorem 6. Suppose
-00
< a < b <
00
P > 0, p
-1
. and q locally lntegra-
ble functions on [a,b) (i.e., in LfocCa,b)). Further suppose q is essentially bounded below. Define T: L 2 Ca,b) + L 2 (a,b) by M[f] = C-l)n(pf(n)) (n) + qf on the domain of the maximal operator
72
RICHARD C. BROWN
T+(M) determined by f such that (pf(n))(i)(a) = 0, i = 0, ... , n - 1. Assume that the minimal operator TO(M) is limit-no Then (1. 3)
b
()
b
2
f ply n 12 + f q Iy I
a
a
b 2 > 110 fly I , a
inf
)1
(J
(Tl ,
0
for all y in D: = {y E L2 (a,b): y(n-l) E AC and the integrals on the reft of (l~) are absolutely convergent}. 1 EquaTity holds Ifty-rs-an eigenfunction corresponding to 110' If 110 ¢ a p ' equari~holds if and only if y = O. But there is a sequence such that IIYkl1 = 1 and 0. f bpl'y(kn ) I 2 + fb qly k 12 - 11 0 fb ly k 12 ~ ~ a a a Certainly Theorem 6 reminds us of Proposition 1. is "what is L?"
The question here
One purpose of this paper will be to give an extension of Proposition 1 which is adequate for Theorems 3-6 and other inequali ties as well. We proceed to outline the contents of the paper. The desired extension of Proposition 1 - "Proposition la" - is presented in Section 2 using a theory of linear relations in Hilbert space developed in recent years by Cgddington [15),[16) and also earlier by von Neumann [27), Krasno~erskfi[25), and Arens [51. The proofs of Theorems 3-6 and of certain additional corollaries will be given in Section 3. Section 4 discusses some extensions of the theory to inequalities with interior point boundary conditions, andsketches the relation of some of our results to those of others. The paper is intended to be self-contained "almost everywhere" in that the significant arguments are sketched in some detail or in the occasional instance where this is not possible full references are given. (The only exception will be Theorem 8, Section 4.) 2. THE SPECTRAL THEORY OF LINEAR RELATIONS Let H, H' be complex Hilbert spaces. A linear relation L is a setvalued mapping on DeL) c H to H' whose graph GeL) is a subspace of H x H'. (We find i t useful to distinguish between Land G(L) although this need not be done. One can identify the relation with its graph and speak directly of subspaces as is done for example in [15) or [16).) L is closed if and only i f G(L) is closed in the usual norm topolo£y of H x H'. L is normally solvable if it is both closed and has closed range. For a E D(L) the image set in R(L) will be denoted by L(a); an arbitrary member of this set will be signified by La. We define IIL(a) II by dist(La + L(O)): = inf{IILa + yll: y E L(O)}, i.e., as the norm of
WIRTINGER AND DIRICHLET INEQUALITIES AND SPECTRAL THEOR Y
73
an element in H'/L(O). Supposing L is closed, L(O) is a closed subspace of R(L) and S E L(a) if and only if S ~ a mod L(O). The nullspace N(t) of L: = L-lL(O) ~ {a E D(L): (a,O) E G(L)}. Given relations L,M we define LoM such that G(LoM): = {(a,S): (a,y) E G(M); (y,S) E G(L)}. The adjoint L* of L has graph ((a,S): [y,a] - [x,S] =O,II(x,y) E G(L)}. Clearly L*(O) = D(L)·J.··. Let L be defined in H. Then A E pel) if (L - AI)-l is a bounded operator from H/L(O) to H. o(L) is the complement of pel). A E opeL) and ~ is an eigenfunction corresponding to A if (~, A~) E G(L), equivalently if A~ = L~ mod L(O). There are close parallels ~etween the adjoint and spectral theory of operators and that of relations. For instance the Fredholm alternatives and closed range theorem are true in both cases (cf. [5 ],[16) , [27) for detai Is) . We now give a generalization of Proposition 1 adequate for inequalities on nondense domains. Proposi tiOH la. Let H, H' be Hilbert spaces and L: H ..,. H' a nondensely deflned normally solvable operator. Then L*L is a s.a. ii:ClTIiiaTIy solvable relation and IIY II 2 ].1(jl/2IILy II where ]..10: = inf o(L*L) . .!.i]..lo E 0p(L*L) equality is attained at 1/1 if and only if 1/1 is an eigenfunction of L*L. .!.i].10 t 0p(L*L) then equality holds if and only if y = 0, --but ----there exists -a sequence .¢n E D(L*L) ----- -- --- ---- - wi th II ¢n II
= 1
such that lim II Un 112 - ].1~l
=
0 as n ..,.
00.
L- l is defined and bounded by the closed graph theorem. Let ~orthogonal projection onto R(L). Then IIL-lpil = Ilvlll and L-lp maps Il' onto D(L). Consider(L-lp)*. Since O:-lp)* = (L-lp2)* = P(L-lp)*, R(L-lp)* c ReL). Let y E D(L) and -1 * -1 * * z E H. Then,[(L P) z, Ly] - [z,y] = 0 so that ((L P) z,z) E G(L ); thus (L -1 1') * maps H into D(L * ). Set T: = L-1 pel -1 P) * . T is s.a. Set S: {(Tz, z + y): z E H' , Y E L* (D)}. Routine computations show that S L*L and that S is s.a. Thus L*L is s.a. We next show o(L*L) = o(T)-l. This means ]..10 is real and positive since T is s.a. Let Q be orthogonal projection on D(L). We claim that T (Q L*L)-l. To see this let y E DeL) and z E H. Then ([T Q L*Ly,z]) = [(L-lp)*Q L*Ly, (L-lp)*z] = [L*Ly, Q L-l(L-lp)*z] = [Ly, (L-lp)*z] Proof.
[y,z], so that y = T Q L*L),. I t follows that oCT) = o(Q L*L)-l. Let ]..I be a complex number and Z E D(L). It is easily checked that II(L * L z - ]..I z) /L * (0) II = II Q L*Lz - ]..IZ II. This fact implies from our definitions that p(L*L) = p(Q L*L) and that o(L*L) = o(Q L*L). Consequently oCT) = o(L*L)-l. To complete the proof, we observe that
=
74
110
RICHARD C. BROWN
- Yz
i1.
= IlL·· 1 II = IIL- 1 pil =
k
I!TI12~
0/T)2. But Further since o(L*L)
ll~l E oCT) <=> llO E o(L*L).
>
0,
llC/ sup 0(T) <=> 110 = inf o(L"'L). Finally, the statements concerniEg equality follow from standard theory (cf. [24], p.234). Corollary 1. Let the hypotheses of Proposition la be satisfied. Suppose also L has ~ compact partial inverse. Then,110 is the least positive eigenvalue of L*L. ~ality is attained by ~ E DeL) if and only i f ~ is an eigenfunction of L *1. Corollary 2.
Suppose L satisfies the hypotheses of Corollary 1. L on Dn: = {y E DeL): [y,E;i) = 0, i =1, ... , n - l } where {(.} are the first n - 1 eigenfunctions of L*L. Then
IJeTII1e---r:;c ~
IlL n II
-=" n-r-where" is n1
the n
th
-*
.
eIgenvalue of L L.
Proof. It can be shown Ccf. [11)) that G(LI~) = {(y, L*y + 1jJ): [y, t:il = 0, 1jJ = L c j i;i} where the c i are arbitrary compl~x parameters. Hence the eigenvalue problem is L*Ly = "y + l)!; [y, E;i 1 = 0, i = l, ... ,n - 1. This, however, implies l)! = 0, so that standard theory applies to show that 110 = "n· 3.
APPLICATIONS
We now show how PropOSition 1 or la applies to the theorems of Section 1. With the exception of Theorem 6 the fact that a given L is normally so~vable and has a compact inverse as well as the structure of L can be read off from theory in [10]or[14). Proof of Theorem 3. Define L by y' on 1 2 211 D: = {y E W ' (0,211): yeO) = y(211); (, y dt = O. Then, l G(L*) = {(y, -y' + ¢): y E W ,2(0,211): yeO) = y(211); ¢ an arbitrary complex parameter}. By Corollary 1 the best constant in Wirtinger's inequality is 1 for -y " y (0) (3.1)
y' (0) f211 y
a
"y
+
¢
y (2iT) , Y (211) 0
for some complex ¢. Integration and use of the boundary conditions in (3.1) shows that ¢ = o. Therefore, 1 = 1 with an eigenmanifold spanned by sin t and cos t.
75
WIRTINGHR AND DIRICHLET INEQUALITIES AND SPECTRAL THHOR Y
2 Proof of Theorem 4. Define L by v" on the subspace of W ,2(0,'Tf) satIsfYIng the boundary condition (3.1). L* is given by y 1--+ y" + ¢ with y' (0) = y' (11). y (i v) y'( 0) Y (iii) (0) 11 J y
The eigenvalue problem is icy + ¢ y ' C1I ) Y (ii i) (11)
O.
° The rest of the proof parallels that of Theorem 3. Proof of Theorem 5.
This is an immediate application of Corollary
~
Proof of Theorem 6.
Here Proposition 1 is sufficient but L needs to be carefully defined. Define L: L 2 (a,b) ~ LZ(a,b) x LZ(a,b) by Pl/zy(n) ) , y (
y I---'"
( (q+d)
D
liZ y
where d is such that q + d > ( > 0. Clearly L is densely defined and 1-1. It is straightfor~ard to show that L is closed. Further . 1/2 (n) 1/2 L has closed range. For If p Yk -7 U and (q + d) Yk -7 V our
-liZ
choice of d guarantees that Yk -7 v(q + d) . But since the operator y I---'" pl/Zy(n) on = {y ( L 2 (a,b): y(n-l) (AC; p1/2y(n)}
D:
is closed (this follows by the hypothesis on p), = u. Moreover (q + d)1/2(v(q + d)-l/2)= v
p1/2(v(q + d)-l/2)(n)
so that (u,v) (R(L). Define L+: LZ(a,b) x LZ(a.b) -7 LZ(a,b) by +( ) -, n 1I Z (n) liZ L Z I' Z Z =: l -1) (J1 Z 1) + (q + d) Z on l D*: = {(zl'zZ): ( p /2 Z1 ln-l)( AC;(pl / 2 z1 )(i (a) = 0, i = 0, ... , n - 1, [y, (b') = o}. [[ere [y, ~] (b-) is a form discovered by integrat-
J
zi
ing (-1) n (pl/2 z/n) y by parts. is to show that L+* = 1.
The next step, which is not difficult,
Therefore, L * = ~ L (this is an operator
since L is densely defined.) It turns out further that if OJ * + .. 1 L* L IS . q ( L (a,b), L = L. By ProposItIon, s.a. Furt h er
*
---=F
-+-
L L := L L = L L.
*
+
Now L L c Td:=T + d whence L L c T . d the limit-n condition Tdis s.a. I t follows that L*L
*.
.,
)lod: = inf a(T d )· Because L L IS posItIve )lO,d > 0, Proposition 1 gives
Since we have Td . Let Applying
RICHARD C. BROWN
76
~bpIY'IZ+
(q + d) lyl 2 > )10d llylZ
.
By the spectral mapping theorem )10d - d = )10 = inf 0(T). discussion of equality also follows from Proposition 1. -1
The
1
Corollary 3. Suppose b < 00, p , q E L (a,b). Then Theorem 6 is true provided functions f in DC!) satisfy c;f(n))(i)(a) = (p f(n))CIT(b) = 0, i = 1, ... , n - 1. Moreover, )10 is an eigenvalue and equality in (1.3) is attained CIt an eigenfunction. Proof. We approximate T by sequence of operators Tn such that qn is essentiall'y bounded below and 1!<1 n - qlll -+ O. By a Gronwall inequality argument we show thai: the fundamental solutions of the Tn are equicontinuous and uniformly bounded. This is enough to show -1 -1 via Ascoli's theorem that Tn -+ T uniformly. A theorem of Gohberg and Krein [ZO] and the fact that T as well as the Tn are bounded below will imnly that )10 is a cluster point of the )10,n ()10 ,n : = inf oCTn )). Since (1.3) is true relative to qn,)10 ,n for each T ,it is not difficult using the above facts to show that it holds n relative to p,q~ and )10. For additional details in the case n = 1 see [13], 4.
and also [3] for a different proof.
MULTIPOINT INEQUALITIES AND HISTORICAL REMARKS
In this section we discuss some applications of Propositions lila to inequalities involving multipoint boundary conditions and conclude with a sketch of some alternative approaches to our results. Define Ln : L2 CO,IDlT) -+ L 2 (0,mlT), m > n - 1 by yen) for y E Wn ,2(0,mlT) and satisfying the boundary conditions y(ilT) = 0, i = 1, ... , m - 1, y(2 j )(0) = y(2j) (mlT) = 0, j = 0, ... , [en - 1)/2]. By the theory of [10] or [14],t,n satisfies the conditions of Proposition 1 and Ln * is given by (_l)nz(n) for z E Cn - 2 CO,mlT) n Wn ,2(ilT, (i + l)lT), i = 0, ... , m - 1, and satisfying the same endpoint conditions as Ln. Corollary 1 applies and Ilyll .:5.- )1-ol/2 I1y (n)1I where )10 is the least positive eigenvalue of L*L. More precisely we may prove Theorem 7. Let n
=
Zk.
Then Ilyll.:5.- Ily(n) II on D(L n ).
Eguality holds
if y - sin t. Proof. One checks that sin t satisfies the boundary conditions ana]produces equality. Theorem 2 applies on each interval (ilT, (i + l)lT). Consequently, Theorem 7 is true for k = 1.
WIRTINGER AND DIRICHLET INEQUALITIES AND SPECTRAL THE OR Y
77
Suppose it is true for k = p. By Proposition 1 the best constant 1 is also the square root of the least positive eigenvalue Wo of LZP*LZP • Since this operator is s.a., w;l = II (LZP*L2PSlll. L zP * LZp ~ LZp+l Together these facts imply that the inequality is true for k = P + 1. Zk (b - a)/m Corollary 4. Let yEW (a,b), m ~ yCa + ih) - O,-r-= O, ... ,m. Then II y II ~ (hili) n II y (n) II with eq:Dity at sin lI(t - a) h We do aot !~now if this result can be extended to intermediate values of n (attempts to do so have so far produced subtly flawed proofs). Also the question of nonsmooth extremals seems open. IIowever for n = 3, m = Z we have a result which may be new.
3 Z Theorem 8. Let YEW ' (0.211) slJch that y (0) and y" (0) = y" (ZII) = O. Then
f"
o
with equality if and only if
y (11)
y(211)=0
I y I 2 < / " I p i i)I Z
-
y
0
sin t.
The proof rests on a computation involving Theorem 2 and Proposition la and will be omitted. We close the paper with a few historical comments. An alternative spectral theoretic approach to Theorem 3 has been given in [17]. Here one defines an operator T: y 1---->- -y" on the subspace M of W2 ,2(0,211) satisfying the orthogonality condition J211 y = 0; y and y' also satisfy periodic boundary conditions. This gaarantees that T is defined in M. Thus T is a reduced operator in the sense of Akhiezer and Glazman ([1] ,p. 82) and is s. a. I t has first eigenvalue A = 1 with eigenmanifold spanned by sin t, cos t. The numerical range inequality and integration by parts give Wirtinger's inequality on D(T). The inequality is extended to the larger domain D(L) by an approximation technique. This method however seems difficult to generalize to other Wirtinger-like inequalities, e.g., Theorems 3-5 above. By contrast our method gives the same equations as a calculus of variations approach and thus can be viewed either as a spectral interpretation of this approach or as a rigorous justification of it. Further details and other results are given in [12]. In the past decade much work has also been done on Dirichlet functional inequalities in the case n = 1. See, for example, Bradley and Everitt [7],[8], Amos and Everitt [2 -4], Sears and Wray [28], and Everitt and Wray [18J. Additionall~ material relating to the higher order case but in a different setting can be found in 19] and 122]. The methods and hypotheses of these papers, however,
78
RICHARD C. BROWN
differ from our own. Implicit in much of this work is the discovery that the domain on which the inequality is valid is the domain of the square root of Td . This fact also follows from our approach; indeed D(L) = IX/I7'L) for all the inequalities considered in this paper as is clear from Kato [23], Ch. 6.7 (2.22) p.334. We have also not considered the weight function case here (as is done in [18]). This case produces an inequality of the form
-1
b 2 f plf' 12 + qlfl > a , ,
b 2 f wlfl °a
jJ
where M[f]: = w [- Cpy) + qy] with q > -kw. But such an extension by our approach would be simple. Also~ our method works well for more complicated boundary conditions than considered explicitly here. On the other hand it does not yield inequalities like (1.1) or (1.2) of [18]. For further details and some extensions see [13]. 1.
AC means local absolute continuity in the singular case. REFERENCES
[1] Akhiezer, N.I. and Glazman, I.M., Theory of Linear Operators in Hilbert Space, Vol. I (Ungar, New York, 1961). [2] Amos, R.J. and Everitt, W.N., On a quadratic integral inequality, Proc. Roy. Soc. Edinburgh,Sect. A 78 (1978) 241-256. [3] Amos, R.J. and Everitt, W.N., On integral inequalities associated with ordinary regular differential expressions: Eckhaus, W. and Jager, de F.M., (eds.), Differential Equations and Applications (North-Holland, Amsterdam, 1978). [4] Amos, R.J. and Everitt, W.N., On integral inequalities and compact embeddings associated with ordinary differential expressions, Arch. Rational Mech.Anal. 71 (1979) 15-40. [5] Arens, R., Operational calculus of linear relations, Pacific J. Math. 11 (1961) 9-23. [6] Beckenbach, E.F. and Bellman, R., Inequalities (SpringerVerlag, 1961). [7] Bradley, J.S. and Everitt, W.N., Inequalities associated with regular and singular problems in the calculus of variations, Trans. Alner. Math. Soc. 182 (1973) 303-321. [8] Bradley, J.S. and Everitt, W.N., A singular integral inequality on a bounded interval, Proc. Amer. Math. Soc. 61 (1976) 29-35. [9] Bradley, J.S., Hinton, D.B., and Kauffman, R.M., On the minimization of singular quadratic functionals, preprint. ~O] Brown, R.C., Duality theory for nth order differential operators under Stieltjes boundary conditions II: Nonsmooth coefficients and nonsingular measures, Ann. di Mat. pura ed appl., 105 (1975) 14l-l70. [11] Brown, R.C., Notes on generalized boundary value problems in Banach spaces, I adjoint and extension theory, Pacific J. Math., 85 (1979) 295-322.
lVIRTINGER AND DIRICHLET INEQUALITIES AND SPECTRAL THEOR Y
79
[12] Brown, R.C., Wirtinger's inequality and the spectral theory of linear relations, preprint. [13] Brown, R.C., The minimization of a Dirichlet functional as a problem of operator theory, preprint. [14] Brown, R.C. and Krall, A.M., Adjoints of Stieltjes boundary value problems, Czech. Math. J., 27 (1977) 119-131. [15] Coddington, E. C., Spectral theory of ordinary differential operators, in: Dold, A. and Lckmann, B. (eds.), Spectral Theory and Differential Equations (Lecture Notes in Mathematics #44S, Springer-Verlag, Berlin, 1975). [16] Coddington, E.C., Adjoint subspaces in Banach spaces with applications to ordinary differential subspaces, Ann. di Mat. pura ed appl., llS (197S) I-lIS. [17] Everitt, W.N., Spectral theory of the Wirtinger inequality, in: Dold, A. and Eckmann, B. (eds.), Ordinary and Partial Differential Equations, Dundee 1976 (Lecture Notes in Mathematics #564, Springer-Verlag, Berlin, 1976). [18] Everitt, W.N. and Wray, S.D., A singular spectral identity and equality involving the Dirichlet functional, preprint. [19] Fan, K., Taussky, 0., and Todd, J., Discrete analogs of inequalities of Wirtinger, Monatschefte fUr Mathematik 59 (195~ 73-90. [20] Gohberg, I.C. and Krein, M.G., Introduction to the Theory of Linear Non-selfadjoint Operators (Translations of Mathematical Monographs Vol. IS, American Mathematical Society, Rhode ISland, 1969). [21] Hardy, G.H., Littlewood, J.E., and Palya, G., Inequalities (Cambridge University Press, 1967). [22] Hinton, D.B., Eigenfunction expansions and spectral matrices of singular differential operators, Proc. Roy. Soc. Edinbufgh, Sect. A. 80 (1978) 289-30S. [23] Kato, T., Perturbation Theory for Linear Operators (SpringerVerlag, Berlin, 1966). [24] Krall, A.M., Linear Methods of Applied Analysis, (Addison-Wesley, Reading, Mass. 1973). [25] Krasnose~skiI , M.A., On the extension of Hermetian operators with a nondense domain of definition, Doklady Akad. Nauk SSR (N.S.) 59 (1948) 13-16 (Russian). [26] Mitrinovic', D.S., Analytic Inequalities (Springer-Verlag, Berlin, 1970). [27] Neumann, J. von, Functional Operators, 1., Annals of Math. Studies, No. 21 (Princeton University Press, Princeton, 1950). [28] Sears, D.B. and Wray, S.D., An inequality of C.R. Putnam involving a Dirichlet functional, Proc. Roy Soc. Edinburgh, Sect. A. IS (1975/76) 199-207.
This page intentionally left blank
This Page Intentionally Left Blank
Spectral Theory of Differential Operators I. W. Knowles and R. T. Lewis (eds.) © North-Holland Publishing Company. 1981
A SURVEY OF SOME RECENT RESULTS IN TRANSMUTATION
Robert Carroll University of Illinois at Champaign-Urbana
I. Introduction. This is a very abbreviated survey of some work done in the past few years on the theme of transmutation (Sections 5 and 6 represent new material).
We shall omit most references, for brevity (they can usually be found
in the bibliographies to our papers) and all of this material will appear organized together in a new book [ 101 which we are preparing now.
We consider only
second order differential operators of the form Qu = (~Qu')' /~Q here (plus suitable perturbations) under two kinds of hypotheses:
(A) Q is modeled on the radial
Laplace-Beltrami operator in a noncompact rank one symmetric space (~Q
+ x or sh + x ch + x for example) or (B) ~Q E c , 0
2m l
2a l
[0,00),
~Q ~ ~Q(oo)
sh
, set Q(D) = Q(D)
+
{~Q("'/~Q)'}' + P~'"
28 l
l
2m+l
h P 2 were P Q
Q
1'2 l'1m uA' /Au Q
Q
as x ~
00
- ~ Q < A < 00 on We
'*'" with formal adjoint Q
(the nonselfadjoint formulation is deliberate and useful).
Here P and
0 are
P = 0)
Q (B: P + 0)
We
if OB = BP acting on suit-
of the form above and B will usually be an inte-
gral operator with distribution kernel. ized translation when
or
< a <
rapidly (such hypotheses (B) arise in many applications).
say that an operator B transmutes Pinto able objects.
x
The theme of transmutation (and general-
has played an important role in the study of partial
differential equations, integral transforms, operational calculus, spectral theory and eigenfunction expansions, fractional integral operators, special functions, inverse Sturm-Liouville theory, quantum scattering theory, etc.
2. General ideas and Parseva1 formulas.
81
In [1;2;3;41 we constructed a
82
ROBERT CARROLL
framework of spaces and maps linking various transforms arising from 1
gether with transmutations Band B = B- . .
sat~sfy
Q 1; Dx"'.\ (0) =
AQ 2Q Q Qip\ = -\ ip\; 'P A(0)
P and Q to-
We let the "spherical functions" ip~
° and set [l.\ (x) Q
Q = AQ(x)ip:\ (x).
Define
for suitable f, Qf(\) = f(\) = ~oo f(x)[l~(x)dX and KF(x) = ~oo F(\)ip~(X)dWQ(\) where
~Q(:\)d:\ is the spectral measure associated with Q; by
dWQ(A) = dA/2nicQ(A)i2 =
f~ = Q-l.
[19;21;26] tions" times
~A (x) ~(x)
'V
Here
ip~(x) = CQ(:\)~A (x) where the "Jost solu-
Qp
= _.\2", and are asymptotic to exp(±i:\-PQ)x as x + 00 (some-
satisfy
A-Y,( ) iAx)
Q x e
with o(x) = ([l~(x),l)w and if
.
-
0Q(x) = O(x)/AQ(x) then 0Q(.\) = 1.
p
p
Similarly for P as above one has "'.\' 0.\ =
Apip~ with Ul~(x),l)v = 6(x), and we write P and n for the corresponding transforms.
One can deal with more general operators
P and
Q and spectral pairings
based on the generalized spectral function of [29]. One way to transmute is via (*) P(D )ip(x,y) = Q(D )'P(x,y) with ip(x,O) = x
f(x) and ip (x,O) = 0.
y
Here one extends data f given on [0,00) to be even on (_00,00).
y
Then assuming a unique solution we write ip(O,y) = Bf(y) and this defines a certain transmutation B (QB = BP). TYf(x) = U(x,y) (T Y x x
When P = Q, 'P(x,y) is the generalized translation
P) and similarly SYBf(n) = V(y,n) (SY'V Q).
'V
n
Define now
n
Q
p
B(y,x) =
([l~(x),ip~(y)v; y(x,y) = (ipA(X),[l.\(y)w; ip(x,y) = (8(y,O,U(x,O); and
I/I(x,y)
P Q P Q (y(x,n),V(y,n).· Theorem 2.1. Given general P and Q, 'P\, 'P , [l.\, [lA' A
A
A
w, and v as above, and uniqueness theorems for all differential equations, then ip = 1/1 satisfies (*) with ip(O,y) = Bf(y) and the transmutation B (characterized by
Bip~ = 'P~) has kernel B(y,x) (i.e. Bf(y) where KF(x) =
('P~(X),F(A)V'
(S(y,x),f(x)) and
Further B-
1
n=
p-
l
with B
KP
= B has kernel y(x,y) (i.e. Bg(x)
(y(x,y),g(y)) and K = Q-l with B = PQ where PG(y) = ('P~(y),G(A)w . • The transmutation constructed via (*) does not use any spectral pairings.
The generalized
translations TY and SY have kernels S(x,y,() and y(x,y,n) given by S(x,y,() = x
x
00 P P P 00 Q Q Q fa 'P.\(x)'P.\(y)[l.\(Odv and y(x,y,n)= fa ip.\(x)",:\(y)£\(n)dw .• Example 2.2. In a
typical situation AQ(x) = x
2m+l
Q ny -m Q one has ipA (x) = 2 (m+l) (.\x) Jm(.\x); [l.\ (x)
x2m+l'P~(x); and ~(:\) = c; :\2m+l where c = 1/2ny(m+l). m
For this Q = Q = Q and m
83
A SURVEY OF SOME RLiCENT RfiSULTS IN ]'RiL\iSMUL1TION
P ~
P
0 (y,x) Q
Q as above with (dv ~ (2/TI)d\ here) m 2 2 m-lz / 2r(m+l)(y -x)+ / TIr(m+lz). For yQ(x,y) one can write
D2 we have a transmutation BQ: D2 Q (y),Cos AX)V
~ (~\
~
->
Y~ ~
2
(x _/)n-m-3/2 dy where
Y: X(Dx/X)n~ g(y)y2m+l.
~
for -!z < m < n-lz and suitable g, (yQ(x,y)g(y)
n l 11T/2 - r(m+1)f'(n-m-lz).
We will say more about SQ
and YQ later. Let us briefly indicate some techniques of studying Parseval formulas
r 22;29]
in our framework which are based on b Let fey)
(B *O(y)
~
~
(cf. also [12;13;26]). -
V and g(x)
(y(x,y),f(x)
(B *g)(x)
~
~
(S(y,x),g(y).
Qf(A) ~ Pf(A) and Pg(A) ~ Qg(\) where Pf(A) ~ ( f(x) ,~~ (x) -k
~
Note also that B ~
QP and B*
~
PQ where QG(y)
Lemma~.
Then
and Qg(\) ~ ( g(y), (G(A),Q\Q(y)w and PF(x)
~
P
(F(A),Q\ (x)v. - Now one wants to determine a Parseval formula of the form
-~ -!2 (II) (R,PfPg\ ~ (6 p f,~p g) for Pu ~ (~pu')'/~p - q(x)u where q is a suitable po-
tential and R is a generalized spectral function. m
First let B: P
m
change in notation - here v y
form (t) Txop(x)
~
(R
"u
~
D2 ~ Q
+
Pm and w
P
w
~ P
P
(A)'~A (x)~\
0,
Q).
Q (cf. Example 2.2 but note the One tries to find rYe (x) in the x P
(Y)w ,.here dw
(~;lzf,~;\) ~ (g(y),(T~Op(x),f(x)) ~ Y
ul
~
(2/TI)d\.
KP to get (Bop)(y) ~ KPPR
w
P
w
Now B
Operate on this with B
~
Thus RW(\)
BQ from Example 2.2 so Y Q
~
Setting
~ kR w (since K- l ~ PPQ and Q-l ~ k); this holds when-
~~ (no spectral considerations).
Cos Ay dy.
Then for suitable f,g
(RU10),PfPg)w which reduces to (II).
0 now in (t) we have apex) ~ (R (A)'~A (x)w ~ f-lR.
ever B~~
be the
m
We connect P
model of Example 2.2. transmutations.
To illustrate let P
~
ker B.
Some nontrivial but rou-
tine calculations with distributions show (as is apparent on other grounds) that
Now having "discovered" Ro we connect P Again try (1") where TY x
"u
are as in Example 2.2).
2
(m -~)/x
2
and x
m+lz
"u
1<
~;).
P ~ P m
q now and w
P
~
P
q and Q
m
Q
m
A transmutation B:
P+
~
R dA and
P (thus dw = m 0 Consider P and Q where xm+lzp ~ = P {xm+lz~} "u
Q
m
(P
=
Pm· ,n Y
Q
A'
QQ
A
D2 _
m
Q corresponds to a transmutation
84
ROBERT CARROLL
~
B: P + Q and we write B where L(y,x) = y
=
-m-~~
=
Q
sequently, given Bp,
=~,
A
A
.
w one has formally R
0
w w R + R q
0
y
L(y,x)f(x)dx + fey)
o
=
QBO
m
) in a certain space W1 , (R,H} O _h ( supp h compact m
o
f
c 2 A2m+lRw
=
A
/" h(y)l(y)dy (H"v PfPg).
=
UJ
f=f(Y)im+l~9(y)dy + 1; R
=
A
=
(x + y) with Bf(y)
Assume (t) with R = QBo p ' We obtain Bop(Y) = 2m+l lim L(y,x)/x as x + 0 which is well defined. Con-
L(y,x)x
p
H
Bx
m~...-z
, op(y) + £(y) where £(y)
l(y)}~9(y)y2m+ldy
-m-!z'" m~"2
Y
=
P
~
f
00
0
R + R. q
0
=
{op(Y) + Here, for
lim h(y)y-2m-l + y+O
Therefore passing from (t) to (II) again - Theorem 2.4.
For suitable f,g of compact support and suitable q(x) one has a Parseval formula (II) with R E (WI), as indicated . • m
3. The Gelfand-Levitan and Mar~enko (GL and M) equations and connection formulas for special functions.
The GL and M equations arise in quantum scatter-
ing theory as a vital part of the machinery used in recovering the potential.
In
[20] Fadeev gives a unified approach to these equations for the Schrodinger equation via certain transmutation operators and the link between the GL and M equations is shown to be a certain transmutation U.
In [5;6;7] we generalized this
procedure both conceptually and technically to the context of harmonic analysis in symmetric spaces.
Our generalization
B (or B = B- 1 )
complementary triangularity properties to B:
~~
of U turns out to have
~~ and this leads to an abstract
+
derivation of (and transmutation meaning for) numerous connection formulas for special functions involving Riemann-Liouville and Wey1 type fractional integrals (cf. [12;13] - see also [26] and Section 6).
= ~~ and setting W(A) = IC (A)/C p (A)1 2 (so that Q
More specifically, dV(A)
W(A)dw(A»
=
our generalization of
nQ.
B= B-
has the form B
=
-
and has the form B ker Band y Ap(x)AQ (y)
p
Q
= (~A
{X),QA{Y»V
- Another important fact involving
B concerns
0 for y > x; y{x,y)
P
W(A)~A
p Q (QA{X)'~A(Y)}W
ker B) have the triangularity properties B(y,x) =
) where - Theorem 3.1.
KP and the distribution kernels (6 -
y{x,y)
B
~~
The transmutation B: Q + P is characterized by 1
--1
Uis B (B
=
=
-1
Ap (x)AQ(y)S(y,x)
the function
-1
=
0 for x
>
y.
~~(Y)/CQ(-A), which in
85
A SURVEY OF SOME RECENT RESULTS [N TRANSMUTATION
certain 'vays is a natural generalization of 2e
iAy
Thus, writing subscripts Q on
.
all kernels and operators when P ~ D2, • Theorem 3.2. Under suitable hypotheses
Theorems of this type and triangularity results are known for special cases (Bessel functions, etc.). the correct transmutation
What seems striking is that once we have isolated
B or B,
based on scattering theory arguments involving
the GL and M equations, then abstract proofs of Theorems 3.1 and 3.2 can be proX
Now write W ~ KW(A)Q and
vided, furnishing information about special functions. for W(A) ~ l/W(A) set WX ~ nW(A)P. BWX
=
B (i.e.
Then the generalized extended GL equation is
= I) when written out in terms of kernels.
BBWX
r 7] and recall
a formula in spherical functions (cf. r,(y,x)
=
0 for y
o
(11WP)}(x» ( ~ ~ (x)
=
S(y,x) where W(y,x)
and this can be written
,~~ (y) >U).
fow
=
~p(X)T~W(X)
=
-
~ Q and W differed by a D2) we can write a
Using the situation of Example 2.2 for Q (p
x sgnx (x2_/)~m-3/2
Ymy-2m{It;I-2m-2
=
equation (relative to P
(ym =
=
, _
Q
=
AQZQ or BQ
'-1:::,
AQ~QWQ ~Q
Q From above (!\(y)/CQ(-A)
= F~Q so that SQ(y,x)
f~ '1'Q (x-t;) f (x)dx and then EQ BQW
=
=
=
BJQ
o
=
=
'
Define ZQf(t;)
=
~).
Write iQf
Define SeA)
= =
~Q *
f and consider BQZQ
CQ(A)/CQ(-A) and set Set) _ '-1:::, ~QWQ ~Qf =
(I+S)f
Hence - Theorem 3.4. The M equation (for D2 and
AQ(HS) or AQ(y,X) +
f~ BQ(y,O~Q(t;-x)dt:
00
'iAx AQ{e }(y)
2fy SQ(y,x)exp(iAx)dx.
Then for functions f defined on [0,00),
where Sf(y) = foo S(y+x)f(x)dx. q) is
=
=
=
We adj oin BQ~ ~ BQ (GL equation) to get
AQZQ'
AQZQW;l (WQ
fS.
Now for the Mar~enko
f~ ~Q(X-t;)AQ(y,t;)d(.
(this is the M equation).
(1/2n)fS so S(-A)
(/_t;2):-~}(x).
1,
r(rn+l)/ITIr(m+~»
2 Q D and Q) we generalize [20] and set $A (y)
'iAx (AQ(y,x),e > ~ FAQ(y,).
Set ~/CQ(-A)
yand
(~~(O,~i(Y»J'p(OT~(Odt;
typical version of the generalized GL equation in the form
~
~ >
(~(x),~~(y) >w (W(x) ~
In [5;6] we dealt mainly with transmutations D2 factor of 2.
= 0 for
x). - Theorem 3.3. The generalized extended GL equation is
>
B(y,~)W(~,x)d~
fW
S(y,~)
We display this as
f; AQ(y,t;)S(t;+x)dt: ~ 0 for y
f; AQ(y,OS(E;+x)dC
-
<
x; for y
>
x
86
ROBHRT CARROLL
4. Inverse problems and integral equations. we considered a problem in geophysics: p(x)v
;
tt
"
(p/~) 2(0)6(t); with readout v(t,O) ;
vx(t,O) ;
shear modulus
~
are unknown.
one cannot do, but a product
In [14;151, with F. Santosa,
{~(x)vx}x;
v(t,x) ;
° for
<
0;
get), where the density p and the
The inverse problem is to determine p and P~
t
~,
which
as a function of a bound variable y (dy/dx
'"
(r /~) 2) can be determined using methods derived from quantum scattering theory and the transmutation techniques above.
; ° with (p~)
We reduce the equation to (A~ ) /A + k2~ .
y y
~ (k,O) ; 1 via Fourier transform and a change of variables (A(y) y
"
Then spherical functions ~(k,y) and Jost solutions ~(±k,y) are con-
2(y».
structed by methods of integral equations and various analyticity and growth proWe have hypotheses of type (B) and set q(y) ; -A'/A, A
perties are determined.
(~(k,y)
A(OO) , etc.
One has a spectral function dv(k)
; c(k)(k,y) + c(-k)(-k,y»
~(k)dk; dk/2~Aoolc(k)12 and from the formulas
for v one obtains, using contour integration, ~(k)
=
-(2k/nA )f~ g(t)Sin kt dt. o
0
Thus the spectral function ~ is determined by the readout g and one can now derive the appropriate Gelfand-Levitan machinery. define T(y,x) ; foo
Sin kx k
o
Set dv (k)
=
do (k) + (2/1,) dk and
Cos ky doCk) •• Theorem 4.1. Let K(y,x) be the
(unique) solution of the GL equation K(y,x) + T(y,x) = fY K(y,~)T (n,x)dn (x ~ y). ~
o
_h
11
Then q(y) can be recovered from K(y,y) ; 1 - Ao A 2(y) . • In order to deal with a physical problem where complete recovery of geophysical data was possible Santosa and the author considered in [16;17] the following problem.
3 Take an elastic halfspace in R , xl ~ 0,
stratified in the xl direction, with density p(x ) and Lam~ moduli (A,~)(xl)' l One imposes stresses 'li(t,x) ; 6(x ,x )6(t) at the surface xl ; 0 and given dis2 3 placements u
( i ; 1,2,3) we form variables vi(t,x ) ; ff u i dx 2 dx and w(t,x ) l l 3
i
ff x u dx dx . 2 l 2 3
It suffices to work with the equations for vI' v ' and w. 2
introduce variables Yl and Y2 with dYl/dx ; and set Al ;
(p(A+2~»
k
2 with A2 ;
(Py)
k 2.
We
"
(p/A+2~) 2
2
Then Dtv
i
(A.V~)'
/A. where
111
I
denotes
d/dYi (i ; 1,2); after Fourier transform these have the form treated above.
One
87
A SUR Vii Y OF SOME RECENT RES ULTS IN TRANSMUT!l nON
obtains then spectral densities ~.(k) from readouts h.(t) as before and Theorem ~
~
4.1 can be applied to obtain Ai(Yi)' the form Al (yl)dY 2
Dtw
(AIDlw\/A
=
\(O)A
-1
l
=
l
l
This gives a relation between Yl and Y2 of
A (y 2 )dY with Ai known and the w problem now reduces to: 2 2
- B(y )v (y,y ) - D(y )D V (t,y ); ,,,(t,O) 2 l 2 2 l 2 2
(0)h (t); w(t,y ) 2 l
=
0 for t
0 (here Di'v d/dy ). i
<
jet); Dlw(t,O)
=
=
D(y ) is known and l
p(x ) is the only unknown. l
After Fourier
transformation one arrives at an integral equation for B. • Theorem 4.2. B(y ) l satisfies the integral equation f(k)
=
-~= ~l (k,nl)~2(k,n2)B(nl)Al(nl)dnl where
f, ~l' ~2' and Al are known and n 2 is a known function of n l .
Given a solution (p,\,~)
B(y ), together with relations already obtained above, one finds l tions of x
=
as func-
xl' •
Now rewrite the integral equation as (B real valued) F(\) Q _
00
~
{¢A
writes
P (Y)/CQ(-A)}{~\ (y)/cp(-A)}B(Y)~p(y)dy
BQ{2e
iAX
Similarly Bp{2e co
~
}(y) = ~;(Y)/CQ(-A) where SQ(Y'x) = 0 for Y iAX
}(y)
¢~(Y)/Cp(-A) and setting f(t)
=
B(y)G(y,t)dy; G(y,t)
t-
=
~p(Y)fy
this is a Volterra type equation. -1
~p
p
v
(y){h (y)6(x-y) + Kp(x,y)} where
holds when (p,A,W) E f(T)
=
B(T) + f
monotone.
and in [16;17] we solved this.
T(T)
o
c
2
>
x (SQ
One
ker B ). Q
=
F-IF one obtains f(t)
=
~-
Sp(y,s)SQ(y,t-s)ds so G(y,t)
0 for y
=
t and
>
Moreover it is natural to take e.g. Sp(Y,X) -1
v
Kp(x,y)~p
2
(y) E L
for example even with L12
oc
2 B(y)K(y,T)dy where K(y,T) E Ll
loc
'
Such a decomposition
l replaced by C
Then (**) 1
oc
=
and T(T) E C
is strictly
It is also realistic to take f E L2 . • Theorem 4.3. Under the condi-
tions indicated (**) has a unique solution B E Lioc' • One can find various equivalent formulations of the integral equation in decomposing the kernels Sp and SQ in different ways.
This involves a number of
formulas and relations (based in part on the distributional Hilbert transform) which are of interest.
-2AO;(y) of
=
In particular we use rr~Q(A)~;(Y)
Im{~;(y)/cQ(-A)} where e;(y)
(~Q~')' /~Q
=
_A2~
(p
Q
=
=
~Q(oo)e;(Y)/~Q(O),
Re{~;(Y)/CQ(-A)};
e;
being the solution
= 0 here) satisfying e;(O) = 0 and Dxe~(O)
=
-1.
We
88
ROBER T CARROLL
also considered other integral equations with i(y) type kernels (cf. [8]). sider e.g. (1111) FCA)
I; f(y)L:.Q(Y){~(y)/cQ(->-)}dY
=
(cf. also [30]) for which we
have - Theorem 4.4. The solution of (1111) can be written fey) fey)
Con-
=
Y,(Bl-lF) (y) or
(1/2n)/" {F(A) + F(->-)}.p~(y)dA. -
=
o
Similarly an integral equation F(>-)
I;
=
f(y)i(y)dy (assume CQ(-A) is
not known) can be reduced to a Volterra equation if we can compute AQ(y,x) (1/2n)[:
~(y)exp(-iAX)d>- where by known triangularity AQ(y,x)
Setting T(x)
'"
0
very rapidly decreasing q, SeA) ImA > 0 and fey)
=
(1/2TI)!:
=
x.
>
2 _A u and for suitable
=
CQ(A)/CQ(-A) will be analytic in a halfplane
F(A)~A (y)dA.
5. Elliptic transmutation.
The transmutations P
~
by spectral pairings required basically that the spectra of Consider now - Example 5.1. Let P = D2 =
P and
Q = _D
2
2 For Q consider _D W = _AZW with W(O)
with A E [0,00). AX
0 for y
= (1/2n)I'" F(A)exp(-iAx)dA one obtains IX AQ(y,x)f(y)dy = T(x).
Still another approach refers to equations u" - q(x)u
e-
=
Set Qf(A) = f(A) =
Z ZY/TI{x +y2}.
Also
B<{J~
=
.
-
I; T~S(y,E;)f(E;)dt; = Y,f:
=
0).
P For P, <{J A(x)
=
A
eAx with J-\ F(x) P Q U\(X) ,<{JA(y) \
=
<{J~ and since T~f(X)
f(x-y)} we obtain (extending f to be an even f) <{J(x,y)
(T~S(y,E;),f(t;» 2 Z
identical. =
Cosh
1 and take say <{Ji(x)
=
f2~(x)
A A Consider B: P ~ Q with kernel S(y,x)
I; COSAX e-AYdA
P and Qbe
=
~oo f(X)<{Ji(x)dX = Lf(A) (Laplace transform) and KF(x)
(l/ZTIi) I F(A)eAxdA (contour integral - i. e. set AQ (F(A),S\(x»w).
= Q.
Q constructed so far
=
=
=
(Z/TI)
=
Y,{f(x+y) + x
(S(y,O,TE;f(O) =
S(y,x-t;)f(t;)dt; = (y/n)[: f(t;)dt;/
{(x-E;) +y } (y
>
The equation (*) of Section Z for <{J is then an elliptic
equation D;<{J
-DZ<{J and <{J is the solution of a halfplane Dirichlet problem via y
a Poisson integral formula; <{J(x,O) = [(x) is specified and we obtain uniqueness by imposing growth conditions (e.g. <{J bounded).
We will refer to this kind of
situation as elliptic transmutation and give some preliminary results in [9]. recall also that the conjugate harmonic function to <{J is (y > 0) W(x,y) '"
-
2
Z
.
(l/n)[",{(x-i;)f(OdU[ (x-O +y ]) and as y
-+
0, W(x,y)
-+
(l/n){Pf(l/O
*
-
O(x)
We
89
A SURVEY OF SOMli RIiCFNT RESULTS IN TRANSMUTATION
H denotes
-Hf where
the Hilbert transform. -
Take now a model situation (m generalize.
Set Q
(n;(x),.p~(y» A A
V
A
Q = -D
2
>
Q
with.p A(y)
-lz) P = Pm again; the constructions then e
_\y
A
A
and transmute P -.- Q via 8 (y, x)
cm t' (h)m+l h km J (h)e-AYdA = kmYX 2m+l/( x 2+ Y2)m+3/2 were om
2f'(m+3/2)//7Tr(m+l).
TY ," P is known (cL [4;27]) and we have - Theorem 5.2. x m
The function.p (x, y) = ( S (y, 1;), T~f (I;)
2 y
satisfies P (D )'1' = _D 'P and 'I' (x,O) = f(x) m
x
,,,ith.p (x,y) = /" P(y,x,OI;2m+lf(i;)di; where p(y,x,l;) = T~{B(y,I;)i;-2m-l} (i;x)-m f
o
oo
e-ytJ (xt)J (i;t) tdt. -
o
m
m
One makes contact here with generalized axially symmetric potential theory as developed by Weinstein (cf. [34) tions.
and the study of pseudoanalytic func-
Thus, conjugate to the function 'I' above will be a function "'(x,y)
x2m+l foo Q(y,x,Oi;2m+l f (Odi; (Q is constructed below by transmutation) and ('I' ,"')
o
satisfy the generalized Cauchy-Riemann equations "'x Let now
=
x2m+l.p
X~(y) = e-\Y/A so that X~(O) = l/A and DyX~(O) = -1.
y'
'"
y
= _x2m+l.p . x
Define a transmuta-
P Q 2m+ 1 / 2 2 m+'" tion B: P -.- Q with kernel B(y,x) = <S\(x),X\(y»v = r(m+lz)x / 7Tr(m+l)(x +y) 2 v
v
A
v
P
Q
(note B'P\ = X\). - Theorem 5.3. Define G(!;,x,y) = -DyG(i;,x,y) and Q(y,x,l;) = DxG(!;,x,y) is the conjugate kernel.
Qy G
One has
Px G =
and G is a "fundamental solution" with a logarithmic singularity at the point
(y=O,x=!;).
In particular (Qv denotes a Legendre function) one has G(!;,x,y)
(!;x) -m foo J (xt)J (I;t) e -yt d t m m
(l/rr)(C;x) -m-lz~_lz{ (x 2+/+<:2) /21;xL _
The conjugate kernel leads to a generalized Hilbert transform (originally developed by Muckenhoupt-Stein) which has been studied extensively in connection with Erdelyi-Kober operators, etc. (see e.g. I 31;33]).
Thus the conjugate
Hankel transform (cf. [33J) is Hmf(x) = ~~~O ~oo Q(y,x,i;)f(i;)i;2m+ld i;.
Our develop-
ment above gives a transmutational background for such an operator and will lead to other such operators as P varies.
There is also another generalized Hilbert
transform studied by Heywood, Kober, Okikiolu, Dettman, et. al. which is transmutational in nature and is connected to P. m
Thus we will transmute Q = _D
2
to
90
P
ROBER T CARROLL
=
Pm for m
<
-!;; (B: Q -+ P) and y(x,y) A
2r(!;;-m)x
=
-2m.; 2 2 "'-m /rrr(-m)(x +y ) 2 (:t:).
The
corresponding Poisson integral formula is then (sy ~ Q) ~(x,y) = (y(x,n),SYf(n) x n 00 2m 00 2 2 "'-m = !;;[oo y(x,y-n)f(n)dn = cmx [00 f(n)dn/{x +(y-n) } 2 • ~(x,y) is the real part A
-
of a pseudoanalytic function in the halfplane x > 0 with conjugate function
~(x,y)
,00 2 2 !;;-m = c [", (y-n)f(n)dn/{x +(y-n)} and one obtains a generalized Hilbert transform m lim 00 I I-2m in the form Hmf(y) = x -+0 ~(x,y) = c [", sgn(y-n)f(n)dn/ y-n (m < -!;;). m A
It is interesting to note that we can give a spectral formula for the y just constructed using the w pairing.
Thus. Theorem 5.4. The kernel y(x,y) of
(+) can be written y(x,y) = (E~(x),D;(y)w = (1/2rri) f~~(x)eAYdA (contour inteP gral); EA (x)
=
-m Ym(AX) {Km(Ax) - Ym(AX)}, where Km denotes the Struve function,
Y the standard Bessel function, and m 2/S( -m, ') = A'y Ix' '2, { Pm ( Dx ) + A2}~~ A m '
m 2 /rrr(!;;+m)/S(-m,!;;).
Ym =
Pill (D)y X
Q(D)y Y +
Thus for
Ym =
Ymo'(y)/x . •
6. Singular pseudodifferential operators (psdo).
The constructions of
Katrakhov and Kipriyanov [23;24;25] for certain singular psdo involving Q = D2 m
+ «2m+l)!x)D can be reformulated in a more "canonical" way.
In particular this
allows one to deal with a larger class of singular psdo involving Qu
(6 u')'/6 . Q Q
=
Thus let a(x, A) be a "classical" symbol acting by Fourier transform, i. e. A(x,D)u =
00 iAX (1/2rr)[00 e a(x,A)Fu(A)dA (Fu =
(u(~),e
-iA~
) here).
Since one is working with
halfline problems we can take a(x,A) even in x and A; further one can make assumptions of compact support in x with say a(x,A) (cf. [32]).
Thus we are only concerned with a cosine transform FC (F 2
for our standard transmutations BQ: D (1/2rr)Fa(x,A) =
= (l/rr)FCa(x,A) and set
A
-+
Q we have P
~(s,A)
(2/rr)f; f; ~(~ ,A)COS sY ;:'(A)dsdA where;:' A
as before and define formally A(y,Q) Theorem 6.1. Set ~
~oo
0 for IAJ ~ ~ in a standard manner
=
=
Qu and ~
f; ~(~,A)~(~)~~(Y)~(A)dsdA
=
~
Q
(=
=
=
=
FC'
=
2FC) and
Let us write a(~,A)
a(~-A,A) + a(~+A,A).
Then A(y,D)u
Let now BQ = KP and BQ = I-\P be
F Cu. -1
A
_
A-I
= BQA(x,D)B Q with A(y,Q) = BQA(x,D)B Q . R
0
and A(y,Q)u
for Q =
~oo
=
=
0). ~
Then A(y,Q)u
=
f; ~(~,A)~(A)~~(Y)~(A)dsdA
.•
The constructions of [23;24;25] for Q = Q involve working on the m
•
A SURVEY OF SOMF RIiCENl" RFSULTS IN TRANSAfUT·1TION
function {exp(iAx)a(x,A)} with e.g. pression with 'I';(y) and
'i';'
(y) (0'
91
o(y,x) as in Example 2.2 to produce an ex'x,
x
2m
'+1
somewhat contrived and is noncanonical.
m'
=
m+l).
This procedure seems
One fact that is brought out however is
II --1 that for a certain Hilbert space adj oint II, BQ = BQ = BQ which provides a nice apersu of the complementary triangularity properties of the respective kernels. An interesting class of singular psdo is introduced in [25) to which we can contribute some additional information. tions II •
JJ
= KA-JJp
and I
jJ
= nAJJQ,-
Thus working with Q
so that IT
~
define transmuta-
jJ
Theorem 6.2. For a(x,A) as above AjJ (v,Q)u -
dc;d>..
Moreover for JJ = m+l.;;, llm+l.;;g = L{x
position of Erdeflyi-Kober operators
(I:; =
-m-!2
g} where L
0, n = m+l, a
is a com=
S
=
-\,m-\). • The ker-
nel of llm+l.;; is expressed in [25) in a somewhat different way but is equivalent and shows, as does our expression, an interesting decomposition as a sum of a RiemannLiouville and a Weyl type kernel.
REFERENCES
1.
2. 3.
4. 5.
6.
7. 8.
9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20.
R. Carroll, Transmutation and operator differential equations, Notas de Matematica 67, North-Holland, Amsterdam, 1979 , Applicable Anal., 8 (1979), 253-263 -, Applicable Anal., 9 (1979), 291-294 -, Transmutation, generalized translation, and transform theory, I and II, to appear Applicable Anal., to appear 1981 Rocky Mount. Jour. Math., to appear 1981 Some remarks on the generalized Gelfand-Levitan equation, to appear Some inversion theorems of Fourier type, to appear Elliptic transmutation, I, to appear Transmutation, scattering theory, and special functions, NorthHolland, to appear 1982 Some remarks on singular pseudodifferential operators, to appear R. Carroll and J. Gilbert, Proc. Japan Acad., to appear 1981 - and -, Some remarks on transmutation, scattering theory, and special functions, to appear R. Carroll and F. Santosa, Applicable Anal., 11 (1980), 79-81 - and Math. Meth. Appl. Sci., to appear 1981 - and Math. Meth. App1. Sci., to appear 1981 - and Comptes Rendus Acad. Sci. Paris, 503 (1980) K. Chad an and P. Sabatier, Inverse problems in quantum scattering theory, Springer, N.Y., 1977 H. Chebli, Jour. Math. Pures Appl., 58 (1979), 1-19 L. Fadeev, Uspekhi Mat. Nauk, 14 (1959), 57-119
92
21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34.
ROBERT CARROLL
M. F1ensted-Jensen, Ark. Mat., 10 (1972), 143-162 M. Gasymov, Trudy Let. Sk. Spek. Teor. Oper., Izd. Elm, Baku, 1975, pp. 20-45 V. Katrakhov, Sibirsk. Zur. Mat., 21 (1980), 86-97 -, Dok1. Akad. Nauk SSSR, 251 (1980), 567-570 V. Katrakhov and I. Kipriyanov, Mat. Sbornik, 104 (1977), 49-68 T. Koornwinder, Ark. Mat., 13 (1975), 145-159 B. Levitan, Uspekhi Mat. Nauk, 6 (1951), 102-143 J. Lions, Bull. Soc. Math. France, 84 (1956), 9-95 V. Marcenko, Sturm-Liouville operators and their applications, Izd. Nauk. Dumka, Kiev, 1977 M. Mizony, Transformation de Laplace-Jacobi, Sem. Univ. Claude-Bernard, Lyon, 1980 P. Rooney, Canad. Jour. Math., 6 (1972), 1198-1216 F. Treves, Introduction to pseudodifferentia1 and Fourier integral operators, Vol. 1 and 2, Plenum Press, N.Y., 1980 J. Walker, Conjugate Hankel transforms and HP theory, Thesis, Univ. of Texas, 1980 A. Weinstein, Bull. Amer. Math. Soc., 59 (1953), 20-38
Spectral Theory of Differential Operators I.W Knowles and R. T. Lewis (eds.) © North·Hofland Publishing Company, 1981
SPECTRAL THEORY AND UNBOUNDED OBSTACLE SCATTERING (*) J. M. Combes Departement de Math~matiques Universit~ de Toulon et du Var Chateau Saint Michel. 83130 La Garde. France. R. Weder Instituto de Investigaciones en Matematicas Aplicadas y en Sistemas. Universidad Nacional Aut6noma de Mexico. Apartado Postal 20-726. M~xico 20, D. F. The existence and completeness of wave operators, and related questions in spectral theory, are studied in the case of scattering by an unbounded obstacle. We consider Dirichlet, Neumann, and general boundary conditions with very mild assumptions in the regularity of the boundary. INTRODUCTION In this talk we consider the existence and completeness of wave operators in the case of scattering by an unbounded obstacle. These are the most important "foundational" questions in scattering theory, and as is well known they are deeply connected with spectral analysis, particularly with analysis of the continuous spectrum. There is a vast literature in the theory of scattering for the wave equation, or the Schrodinger equation, in the case where the obstacle is a bounded set. All the methods that are available in scattering theory have been applied to this case. Trace class methods have been applied by Birman [1], Kato [2] , Wilcox [3] , Deift and Simon [4] , and Jensen and Kato [5]. For a discussion of these results, and a complete 1i st of references see [6] . A 1a rge amount of work has been done in the application of the Lax-Phillips theory to this problem. The basic reference is the book by Lax and Phillips [7] where a complete list of references is given. Stationary methods, a la Agmon, have been applied by S. T. Kuroda [8]. Eigenfunction expansions have been throughly considered by Wilcox in his monograph [3] , where the energy distributions have also been studied. Finally Enss method has been appl ied by Simon [9] . Much less is known in the case where the obstacle is unbounded. In fact we are only aware of work done by Ramm, [10] and the references quoted there, for a very restricted class of obstacles, namely sets in two dimensions contained in a wedge whose boundary is asymptotic to the boundary of the wedge, and satisfies strong regularity assumptions. Recently Wilcox [11], [12], [13], considered scattering by periodic surfaces (diffraction gratings). The restriction to bounded obstacles is unnatural both from the mathematical and physical point of view. From the mathematical point of view it is the equivalent of restricting the analysis of potential scattering to finite range potentials. This paper is a first attempt to discover a natural class of "short range" obsta-
(*) Research partially supported by CONACYT under grant No. PCCBNAL 790025 93
94
f.M. COMBES and R. WEDER
cles", for which existence and completeness of the wave operators will hold. Our motivation comes partially from the work done in high energy expansions of scattering phase shifts by Buslaev [14] , Mayda and Ralston [15] , Jensen and Kato [5] , and Schrader [16]. These expansions involve invariants of the obstacle 1ike volume, area, and curvature, but no restriction (at least to first order) in boundeness is required, except as a technical assumption in the verification of the trace condition of the Kato-Birman and Krein theorems, on which a great part of the work by the above mentioned authors is based. From the physical point of view the restriction to bounded obstacles is unnatural because unbounded obstacles can appear as idealizations or in real situations. Moreover important physical phenomenon like surface waves can conveniently be described within the framework of unbounded obstacle scattering. In Theorem I we will prove the existence of wave operators by Hormander's method [17] for a very general class of obstacles and boundary conditions, which is certainly close to being optimal. Our main condition for existence is that the set of directions along which the obstacle is unbounded is of meazure zero, plus some mild regularity assumptions on the unbounded part of the boundary. However no regularity assumption is made on the bounded part of the boundary. We consider Dirichlet, Neumann, and general boundary conditions. The most powerful complement to our existence results would be to study completeness and absence of singular continuous spectrum by stationary methods, a la Agmon [ 18]. They have already been used in the bounded case by Kuroda [8]. However we will only consider time dependent methods. Enss method [19] - [22] has been appl ied to completeness in the bounded case by Simon [9]. Since we are ultimately interested in the problem of asymptotic expansions of the phase shifts we will use a method based in a trace condition under which phase shifts can be shown to exist by Krein's method [23]- [ 25] . Although our method is based in the Kato-Birman theory it does not lends to the technical difficulties encountered with this theory when applied to infinite or singular obstacles. Most of the previous work in obstacle scattering with a trace condition is based in the verification that (H+i)-m - (Ho+i)-m, m > 1, or e-tH_e- t Ho is trace class, where Ho and H are respectively -A and -A with boundary conditions. Our results in completeness are based in the verification that f(H)(H-H ) f(H ) is trace class, for a suitable chosen function f. This operator is easi9y ex_ o pressed via the Green's formula, in terms of the obstacle and boundary conditions. As an advantage we do not need to obtain estimates in the resolvent or heat kernels. We obtain the required bounds just by Sobolev imbedding theorem [26]. Moreover our proof does not appeal to the invariance prinCiple of wave operators. In the following section we state our results in existellce and completeness of wave operators. A detailed version will appear in [27] . The asymptotic expansion of phase shifts will be considered elsewhere. II The Results Let Qe' the exterior domain, be an open set. The obstable, Q., is the complement of Qe,Q; = IR n-r2 e • We will not assume that "i is bounded. We ~ill consider the situation where ". has a bounded part contained in a ball of radius R, where no regularity assumptlon will be imposed, and an unbounded part, contained in the complement of the ball, satisfying mild regularity assumptions.
95
SPbCTRAL THEOR Y AND UNBOUNDED ORSTACLb SCA TTERING
We assume that (2.1 ) aile = all e ,l u dll e ,2 ' where dlle 1 is contained in a ball of radius R, and dQ e, 2 is contained in the compl ement of the ba 11. Denote (2.2) Il ={XUl I Ixl>R}. R We will impose our regularity assumptions in dll R, which amounts to require regularity of all 2' We assume that
N e,
where
IlR =k~l Il R ,k Il ,k are open sets with the R Il ,j n Il ,k = cp , j "f k ,
(2.3) property (2.4)
R
R
and where each ~R k has a bounded trace operator on dllR k' We will consider situations where' there is no global trace operator, but there is a trace operator for each of the connected components of Il R. For example an obstable with the set of directions along which it is unbounded of measure zero, or an unbounded surface. Let us take polar coordinates (p,w), PElR+, wcs n- 1 , onlR n . Denote E= {
W
s Sn-1
1
j {
wn }~=1 ' {'\n}~=l ' wn E Sn-1,
wn -> W, An s lR + , An -> and Anwn £: dlle} , (2.5) that is to say E is the closure of the set of directions along which the obstacle is unbounded. Our ~tin condition for existence of wav 0Rerator~ is that E is of measure zero in Sn . Denote by J the operator from L2(lR ) to L (Il e ) given by multiplication by the characteristic function of 11 . And let Ho be the selfadjoint realization of -/'; in LL(JRn). e 00
,
Theorem I Let H be a selfadjoint operator ~n L2(11 ) such that for every ¢sD(H) we have £: H (Il ), JRH¢ = -/';cp in L (Il R). eand 2 R R
¢11l
IIcpIlH (Il ) ,;;; K(IIH cpll + 1I¢1I), (2.6) 2 R for some constant K. Suppos~ that E has measure zero and that HI (OR k) has a
bounded trace operator on L (d~R,k)' for 1 ,;;; k ,;;; N. Finally we assume that for some M > 0
J 31l
(l+lxl )-M dS <
(2.7)
00.
R
Then the wave operators ) = Slim e iHt J e -iHot , W+ ( Ho,H,J exist. t-±oo
(2.8)
Sketch of proof. Let J R be the operator of multiplication by the characteristic function of Il . It is equivalent to prove the existence of the wave operators W+ (H ,H ,J R). ToR do thi s we use Kato' s vers i on [28] of Schechter's theorem [ 29] , [30]~ Let 9 E D(H) and 'l'ED(H o )' then (H<jl, JR'l')-(<jJ,J R Ho'¥)= -
~
k-l
J dllR,k
(;)~ cp ~
- ¢
}n ~)ds,
(2.9)
96
J,M. COMBES and R. WEDER
where we used Green's formula and the existence of trace operator from Hl (~R,k) into L2(3~R k) (by simplicity we do not indicate the trace operators in (2.9)). The coerciveness inequality (2.6) is used to verify that the operators in the right of (2.9) have the properties required by Schechter's theorem. To verify the condition too
J
(2.10) t¢ for a dense set of ¢' s, we use Hormander' s stationary phase arguments [17] and condtion (2.7). Particular cases of Theorem I are H=-6 with Dirichlet, Neumann, and general boundary conditions, see [27] for details. Assumption (2.6) is mild and follows from ell iptic regul arity [31] . We turn not to the problem of completeness. For simplicity we consider the case of an obstacle that is arbitrary inside a ball of radius R,B R, and whose boundary outside the ball is a surface. More general obstacles will be considered elsewhere. We assume (2.11 ) (2.12) d~R,± = aB R,± U L , where aB R = 3B R,+ U 3B R,_' and 3B R,+ n a BR,_ = ¢ . Theorem II Let H be a selfadjoint operator in L2(~ ) such that for every ¢ED(H) we have 2 e ¢I~ E H2(~R)' J R H ¢ = - 6 ¢ in L (rl R). R Suppose that 1I¢II H2i (rl ) 3/2, and that there exists a bounded imbedding from H2m_3/2(arlR,±) into C(arl R,±)' Moreover assume that there exist bounded trace operators, T±, from H2t - l (~R,±)into H2t_3/2(3~R,±) 1 < t < m. Assume that for every ¢ED(H m)
"-t = CJ-¢' an+ +"-t 3n
( 2 . 13 ) • Finally assume that xtH+t)-m is compact where x is the characteristic function of~e n B , and that CJEL (L). R X
E
L
Then the wave operators W+(Ho,H,J), exist, are complete, and are partial isometries with initial space L2 (Rn) and final space Xac(H). Sketch of proof. The proof is based on a new criterium for completeness given in Theorem 2.11 of [27]. The key point is the verification that the operator V = (H(H+i)-m JR(Ho+i)-m _(H+i)-m JRHO(Ho+i)-m, (2.14) is trace class. This is done by using Green's formula, Sobolev imbedding theorem, and condition (2.13). The remaining conditions in Theorem 2.11 of [27] are easily veri fi ed. For more deta i 1s see [ 27] .
SPECTRAL TTIEOR Y A,\TD ('NH()UNDED OBSTACL};" SG-lTTElUNG
97
Theorem II imposer very mild restrictions on the boundary. The essential assumption is 0€L (~). The other hypothesis follows from standard results if the boundary is smooth enough. For examples in the application of Theorem II see [27]. REFERENCES. [1] [2]
M. Birman. Existence Conditions for Wave Operators. Izv. Akad. Nauk SSSR Ser. Mat. 27 (1963) 883-906' )Amer. Mat. Soc. Trans. Servo 2 ~'L (1966) 91-117. -T. Kato. Scattering Theory with two Hilbert Spaces. J. Functional Analysis 1 (1967), 342-369.
[3]
C. H. Wilcox. Scattering Theory for the d'Alembert Equation in Exterior Domains. Lecture Notes in Mathematics. Vol. 442 (Springer New York 1975).
[4]
P. Deift and B. Simon. On the Decoupling of Finite Singularities from the Question of Asymptotic Completeness in Two Body Quantum Systems. J. Functio nal Analysis ~~ (1976) 218-238.
[5]
A. Jensen and T. Kato. Behaviour of the Scattering Phase for Exterior Domains. Commun. Part. Diff. Equations. 301) (1978) 1165-1195.
[6]
M. Reed and B. Simon. Methods of Modern Mathematical Physics Vol. III. Scattering Theory. (Academic Press. New York 1979).
[7]
P.O. Lax. and R. S. Phillips. Scattering Thecry.(Academic Press New York 1967) .
[8 J T. S. Kuroda. Scattering Theory for Differential Operators III. Exterior Problems. Lecture Notes in Mathematics Vol 44 8 227-241. (Springer. New York 1975). -------[9]
B. Simon. Phase Space Analysis of Simple Scattering Systems. Extensions of Some Work of Enss. Duke Mathematical Journal 'L~ 1 (1979) 119-168.
l10]
A. G. Ramm. Some Integral Operators. Differensial 'nye Uranvneniya 6 No.8 (1970) 1439-1452. -
[11]
C. Wilcox. Rayleigh-Bloch Wave Expansions for Diffraction Grantings I. Preprint 1980.
[12]
C. Wilcox. Scattering Theory for Difraction Gratings. Preprint 1980.
[13]
C. Wilcox. Rayleigh-Bloch Wave Expansion for Diffraction Gratings II. Preprint 1980.
[14]
V. S. Buslaev. Scattered Plane Waves Spectral Asymptotics and Trace Formulas in Exterior Domains. Soviet. Mat. Dokl. ~~ (1971) 591-595.
[1~
A. Mayda and J. Ralston. An Analogue of Weyl Theorems for Unbounded Domains I Duke Mathematical Journal 45 1(1978) 183-196; II ibid 45 3(1978) 513-536; III ibid 'L~ 4(1979) 725-731. ---
[16]
R.Schrader. High Energy Behavior for non Relativistic Scattering by External Metrics and Yang-Mills Potentials. z. ~~~i~_~ 'L (1978) 27-36.
98
].M. COMBES and R. WEDER
[17]
L. Hormander. The Existence of Wave. Operators in Scattering Theory. Math. Z. l~~ (1976) 69-91.
[lru
S. Agmon. Spectral Properties of Schr~dinger Operators and Scattering Ann. Scuela Norm. Sup. Pisa ~ 2(1975) 151-218.
[19]
V. Enss. Asymptotic Completeness for Quantum Mechanical Potential Scattering I Short Range Potentials. Commun. Math. Phys. ~l (1978) 285-291.
[20]
V. Enss. II. Singular and Long Range Potentials. Ann. Phys. (N.Y.) 117-132.
[21]
V. Enss. A New Method for Asymptotic Completeness, in: Mathematical Problems in Theoretical Physics. K. Osterwalder editor. Lecture Notes in Physics 116. Springer Berlin 1980.
[2~
V. Enss Geometric Methods in Spectral and Scattering Theory of SchrHdinger Operators.In Rigorous Atomic and Molecular Physics G. Velo and A. S. Wightman eds. Plenum New York 1981.
[23]
M. G. Krein. On The Trace Formula in the Theory of Perturbation Mat. Sb. ~~ (75) (1953) 597-626.
[24]
M. G. Krein. On Perturbation Determinants and a Trace Formula for Unitary and Selfadjoint Operators. Dokl. Akad. Nauk SSSR l~~ (1962), 268-271.
[25]
M. Sh. Birman and M. G. Krein. On the Theory of Wave Operators and Scatterirg Operators. Dokl. Akad. Nauk SSR l~~ (1962) 475-478.
[ 26]
R. A. Adams. Sobo I ev Spaces. (Academi c Press 1975).
[27J
J. M. Combes and R. Weder. New Criterion for Existence and Completeness of Wave Operators and Applications to Scattering by Unbounded Obstacles. To appear in Communications in Partial Differential Equations.
[ 28]
T. Kato. On the Cook-Kuroda Criteri urn for Sca tteri ng Theory. Commun Math.
Physics
~
Theor~
n~
(1979)
(1979) 85-90.
[29]
M. Schechter. A New Criterium for Scattering Theory. Duke Mat. Journal ~~ (1977) 863-877.
[300
M. Schechter. Wave Operators for Pairs of Spaces and the Klein-Gordon Equation. Aequationes Mathematicae ~~ (1980) 38-50.
[31]
J. L. Lions. Problemes aux Limites dans les Equations aux Derivees PartielIes. (Universite de Montreal 1965).
Spectral Theory of Differential Operators I.W. Knowles and R. T. Lewis (eds.) © North·Holiand Publishing Company, 1981
ALMOST PERIODIC SOLUTIONS FOR INFINITE DELAY SYSTEMS C. Corduneanu* Department of Mathematics The University of Texas at Arlington Arlin~ton, Texas U.S.A.
For certain systems with infinite delay, which are representable in the form (1), one gives conditions assuring the existence and uniqueness of an almost periodic solution. The almost periodicity concept used for either solution or free term of the system is chosen among the following ones: a.p. function with absolutely convergent Fourier series; a.p. function in Bohr sense; a.p. function in Besicovitch sense. I NTRODUCTI ON
Let us consider the system of differential equations with infinite delay (1)
x(t) =
J~[dA(S)JX(t-S)
+ fIt),
t
E
R,
where x and f take values in Rn, while A stands for an nxn matrix whose entries are functions with bounded variation on R+. If one assumes fIt) to be an almost periodic function, the problem of existence of almost periodic solutions to (1) is a very natural one. More precisely, we are interested in finding conditions which guarantee the existence of an almost periodic solution, for any fIt) in a given space of almost periodic functions. Since system (1) contains as a particular case the ordinary differential system with constant coefficients x(t)
(2 )
=
Ax(t) + f(t),
t
E
R,
one might presume that a necessary and sufficient condition for the existence of an almost periodic solution is: the spectrum of the operator generated by dAIs) does not contain points of the imaginary axis. The validity of the above assertion in the special case of systems having form (2) is proven, for instance, in [3J. Moreover, when x and f take values in a Banach space, while A in (2) is a bounded operator, the validity of the property is established in [7J. The system (1) contains as particular cases several systems with finite or infinite delay, already considered in the literature. For instance, when A(s) does not have a singular component, the system (1) can be rewritten as (1')
x(t) =
I A.x(t-t.) + JooB(S)X(t-S)dS + fIt),
j=O J
where
j=O,1,2, ... ,
J
0
and
B(s) 99
satisfy the conditions
100
(3)
CONSTANTIN CORDUNEANU
t. > 0,
<
J -
00,
The norm for the matrices Aj and B is supposed to be the operator norm, though conditions (3) are verified for another equivalent norm. The systems with infinite delay of the form (1 ') have been investigated recently by the author [4]. See also [5]. Unfortunately, the systems of the form (1 ') constitute only a scarce occurence in the class of systems having form (1), because it has been recently proved that "most monotone functions are singular" [lOJ. Of course, from (1') one can derive many more particular forms for the system, among them the classical integrodifferential system (1")
x(t) = Ax(t) + J:ooB(t-S)X(S)dS + f(t).
We will discuss in this paper the existence and uniqueness of almost periodic solutions for the above considered systems, relying mainly on the construction of the corresponding Fourier series. SPACES OF ALMOST PERIODIC FUNCTIONS Since we are going to use several concepts of almost periodicity, it is proper to provide a brief description of the spaces of almost periodic functions to be considered. Let us denote by T the set of trigonometric polynomials which can be represented in the form n
(4)
T(t) = L:akexp(iAkt), k=O
with real Ak' and complex a k , the linear space T:
k
Ak"fAj
for
1,2 •...• n.
k"fj,
Several norms can be defined on
n II Till
(5)
(6)
(7)
IIH2 = II TIl =
L \a k \ ;
k=l n
( L \a k \ k=l
sup\T(t)\,
2
1
)'2;
for
t
E
R.
If one completes the linear normed space T with respect to each norm defined above. then the following spaces of almost periodic functions are generated: APabs(R,C) - the space of almost periodic functions with absolutely convergent Fourier series. the norm of each element being the sum of the series of absolute values of its Fourier coefficients; B2(R,C) - the space of almost periodic functions in the Besicovitch sense, with index 2, the norm of each element being given by the formula
/1 LMOS T PERIODIC SOLUTIONS l:OR INHNrrE DEL1 Y SYSTEMS
101
(8)
AP(R,C) - the space of uniformly almost periodic functions (Bohr). It is well known [3J that the elements of AP a bs (R,C) and AP(R,C) are continuous bounded functions on R (even uniformly continuous), while the inclusion APabs(R,C) c AP(R,C) holds true. As far as the space B2(R,C) is concerned, its elements are representable as equivalence classes of locally square integrable functions, each class consisting of functions that differ from each other by a function for which the mean value (8) is zero. Since AP(R,C) c B2(R,C), (8) makes sense for each f E AP(R,C). In this case, 2 l< (M{lfl })2 becomes a norm on AP(R,C). Obviously, by completing AP(R,C) with respect to this norm, one obtains B2 (R,C). It is easy to check that by substituting a trigonometric polynomial for f in (8), one obtains (6). A basic property of the space BZ(R,C) is the validity of Fischer-Riesz type theorem: to any sequence of complex numbers {a k} E £2, and any sequence of distinct reals {A k}, there corresponds a unique element in B2 (R,C), such that its Fourier exponents and coefficients are Ak , and a k respectively. See [2J for the proof of this theorem. Finally, let us remark that the spaces of almost periodic vector valued functions APabs(R,C n ), B2 (R,C n ) or AP(R,C n ), can be defined in the same manner. THE MAIN RESULTS The following existence and uniqueness theorems for almost periodic solutions of the system (1) or (1 ') will be established. Theorem 1. Assume f E APabs(R,C n ), or f E AP(R,C n ) and its Fourier exponents satisfy the condition
If A(s) is an n x n matrix whose entries are complex valued functions with bounded variation on R+, satisfying (10)
det(isI - A(is)) f 0,
s
where I stands for the unit matrix of order n, Stieltjes transform (11 )
A(is)
f:eXP(-itS)dA(t),
R,
E
and A(is)
s
E
R,
is the Fourier-
102
CONSTANTIN CORDUNEANU
then equation (1) has a unique almost periodic solution x(t) (Bohr), Fourier series being absolutely convergent: x(t) ~ APabs(R,C n ).
its
Corollary. Let A(s) be as in Theorem 1, and assume f to be continuous and periodic, of period T > 0, with values in Cn. Then there exists a unique periodic solution of the system (1), of period T. The Fourier series of the solution is absolutely convergent. Indeed, for any periodic function, regardless of the period, the condition (9) is satisfied. The Fourier exponents of a periodic function form an arithmetic progression. Theorem 2. fying (3).
Consider the system (1'), with t j , Aj , j = 1,2, ... , and B satisAssume that f E B2 (R,C n). If the condition (10) is satisfied, and 1 2 ( 1+t ) II B( t) 112~ L (R+) ,
( 12)
then system (1') has a unique B2-almost periodic solution. Remark.
The Fourier-Stieltjes transform A(is) is now
( 13)
A(is) =
I A. exp(-it.s)
j=O
J
J
+
JooB(t)eXP(-itS)dt, 0
s
E
R.
From Theorems 1 and 2, one can see that an appropriate kind of almost periodicity for the function f implies the same, or another kind of almost periodicity for the solution. Nevertheless, the following problem is still open: does f ~ AP(R,C n) imply the existence of a solution x ~ AP(R,C n)? Another type of problem, concerning almost periodicity of solutions (1), would be the following Bohr-Neugebauer type of problem: prove bounded solution of (1), with f E AP(R,C n ), is also in AP(R,C n ). without using hypothesis (10). See [9] for differential equations in
of the system that any Of course, Banach spaces.
PROOF OF MAIN RESULTS Before we can prove Theorems 1 and 2, we will establish the following Lemma. Consider the system (1), with f(t) = b exp(iAt), b ~ cn, A ~ R. If condition (10) is satisfied, then there exists a unique solution x(t) = hexp(iH), n hE C •
Proof. One proceeds by direct substitution of f and x in the system (1). linear algebraic system from which h has to be determined is (14 )
[iAI - A(iA)]h
=
The
b,
and it has unique solution by virtue of condition (10). Corollary. ( 15)
There exists a constant
K > 0,
depending only upon A(s).
Ih I ~ KI b I·
Moreover, the following estimate holds true:
such that
ALMaS]' PERIODIC SOLU110NS l'OR INHNITE DELA Y SYSTEMS
103
(16)
where the matrix norm is the Euclidean one. The proof follows at once if one takes into account that the entries of the inverse matrix [iAl - A(iA)]-l are rational functions of A, with coefficients that are bounded on the real axis, and the degrees of numerator and denominator are respectively (n-l) and n. Moreover the polynomial in the denominator does not vanish on the real axis, and its leading term is (i:\)n. Proof of Theorem 1. From the Lemma and Corollary, one easily finds out that for each (vector valued) trigonometric polynomial m
f(t) = L bk exp(iAkt),
( 17)
1
there exists a unique solution of (1) that can be represented as m
(18)
x(t)
L hk
exp(iAkt),
1
where coefficients satisfy the following inequality (see (15)): (18) By
1·1 one denotes the Eucl i dean norm for vectors in Cn .
Let us now assume that the function
f
in (1) is an element of APabs(R,C n ):
(20)
If one denotes by fm(t) the trigonometric polynomial which is the sum of the first m terms· in the series of f(t), then there exists a unique solution xm(t) of the system (1), with fm(t) as f(t), representable in the form (18): (21)
with
h , k
k
1,2, ... ,
given by
(22)
As seen above, inequality (19) holds true for any m, m = 1,2, ... , and this n implies the convergence in APabs(R,C ) of the sequence {xm}. Therefore, there exists an element x(t) in APabs(R,C n ) which satisfies the system (1), for f(t) given by (17).
104
Assume now that
CONSTANTIN CORDUNEANU
f
E
AP(R,C n ),
with Fourier exponents such that (9) holds true:
It is obvious that an almost periodic solution of (1), if any such solution exists, should have as Fourier series
(24) where the coefficients hk are given by formulas (22). We assume here that x E AP(R,C n ), because the more general assumption x E B2 (R,C n) does not guarantee the convergence of the integral occurring in the right hand side of (1). But taking (16) into account, we conclude that M > 0 exists, such that (25)
This inequality shows that (26)
I Ihkl
k=l
which implies that x(t) in (24) is an element of APabs(R,C n). Therefore, when f E AP(R,C n ), and its Fourier exponents verify (9), there exists a solution of n (1) which is in APabs(R,C ). Concerning uniqueness of the almost periodic solution constructed above, as an element of AP(R,C n), it is a consequence of the considerations we made above. Remark. The condition (10) appears to be also a necessary condition, if we want system (1) to possess an almost periodic solution for any almost periodic f(t) . Indeed, if condition (10) fails for a certain real A, then choosing f(t) as in the Lemma we have to solve system (14). Since the matrix of the system is singular, it is clear that for some b we do not have a solution. Proof of Theorem 2. Let us assume now that f(t) E B2 (R,C n), its Fourier series being (23). Then consider the B2 (R,C n) - function x(t), whose Fourier series is (24). The coefficients hk are given by (22), with A(is) given by (13). Of course, this is a formal approach to the construction of an almost periodic solutionto(l'). To show that the series in the right hand side of (24) is indeed the Fourier series of an element from B2(R,C n ), it suffices to show that {h k} E £2, i.e. (27)
ALMOST PERIODIC SOLUTIONS FOR INFINITE DELA Y SYSTEMS
105
The validity of condition (27) is an immediate consequence of the Bessel inequality for the Fourier coefficients of the function f(t), and of inequa1ity (15). According to Fischer-Riesz theorem for B2 - almost periodic functions, the series in the right hand side of (24) is indeed the Fourier series of an e1ement in B2 (R,C n). We still have to prove that the right hand side of (1 ') has a meaning when we substitute for x(t) a function be10nging to B2 . Since the mean value defined by (8) is invariant with respect to translations, there results that each x(t-t j ) has the same mean value, which easily leads to the conclusion that an element of B2(R,C n ) is taken by the operator
L A.x(t-t.)
(28)
j=O J
J
into an element of the same space. As far as the integral part in the right hand side of system (1') is concerned, a rather direct calculation shows that it defines an operator taking B2 into itself. Of course, condition (12) must be taken into account. The discussion conducted above terminates the proof of Theorem 2. Remark. Particular cases of systems belonging to the type (1') have been recently investigated in [1]. Namely, the systems considered are of the form (29)
with
x(t)
ax(t) + f:B(S)X(t-S)dS + f(t),
B satisfying (12), and
f(t)
an element of
a
=
Const.,
B2 .
NONLINEAR PERTURBED SYSTEMS Some of the above obtained results can be generalized to nonlinear systems of the form (30) where f(t;x) = (fx)(t) ient function spaces.
x(t) = J:[dA(S)]X(t-S) + f(t;x), is an operator - generally nonlinear - acting on conven-
As a sample result, regarding nonlinearly perturbed systems of the form (30), we state the following: Theorem 3. Assume that the following conditions are verified for system (30): (1) A(s) satisfies condition (10) of Theorem 1; (2) f is an operator taking the space of continuous periodic functions, with period T > 0 and values in Cn , into itself, and satisfies the following Lipschitz condition: (31)
I fx - fy I ~ LI x - y I '
106
CONSTANTIN CORDUNEANU
where the norm is the supremum norm in the space of periodic functions. If L is small enough, then the system (30) has a unique periodic solution of period T. The proof can be easily carried out, relying on the Corollary of Theorem 1, and the Banach fixed pOint theorem. First, an inequality of the form Ixl < Klfl has to be established, where x represents the unique periodic solution, of period T, to the linear system (1). Then one can use the closed graph theorem to prove the continuity of the operator f + x. Remark. The problem of existence of periodic solutions to the system (1), under different assumptions, has been discussed in detail in [6J. REFERENCES [1 J Alexiades, V., Almost periodic solutions of an integrodifferential system
with infinite delay (to appear). Besicovitch, A. S., Almost Periodic Functions (Dover Publications, Inc., New York, 1954). [3J Corduneanu, C., Almost Periodic Functions (John Wiley & Sons, Inc., New York, 1968). [4J Corduneanu, C., Recent contributions to the theory of differential systems with infinite delay (Institut de Mathematiques, Universite de Louvain, Vander, Louvain, 1976). [5J Corduneanu, C. and Lakshmikantham, V., Equations with infinite delay: a survey, J. Nonlinear Analysis 4 (1980) 831-877. [6J Cushing, J. M., Integro-Differential Equations and Delay Models in Population Dynamics (Springer, Berlin, 1977). [7J Daleckii, Yu. C. and Krein, M. G., Stability of Solutions of Differential Equations in Banach Spaces (AMS Translations, Providence, 1974). [8] Hino, Y., Almost periodic solutions of functional differential equations with infinite retardation, Tohoku Math. J. 32 (1980) 525-530. [9J Zaidman, S., Solutions presque periodiques des equations differentielles abstraites, Enseignement Mathematique 24 (1978) 87-110. [lOJ Zamfirescu, T.• Most monotone functions are singular, Amer. Math. Monthly (1981) 47-49. [2J
* Research partially supported by U. S. Army Research Grant No. DAAG29-80-C-0060.
Spectral Theory of Differential Operators I.W. Knowles and R. T. Lewis reds.} © North-Holland Publishing Company, 1981
A SCHR~DINGER OPERATOR WITH AN OSCILLATING POTENTIAL
Allen Devinatz l Department of Mathematics Northwestern University Evanston, IL 60201 Peter Rej to 2 School of Mathematics University of Minnesota Minneapolis, MN 55455 A relatively large amount of excellent work has been done during the past decade on general spectral and scattering theories for Schrodinger operators with long range oscillating potentials. However, most of these works do not include operators of the form H = -6
+ c sin br + Vex), r
where V is a short range potential. When 17 is radially symmetric, the problem has been successfully dealt with in recent years. On the other hand when V is not radially symmetric only one recent paper deals with these operators, but only for high energy values. In this paper we shall consider the spectral theory for this specific operator and compare our results with the previously mentioned paper.
§l
INTRODUCTION
In this paper we shall consider the problem of the location of the absolutely continuous spectrum of the self-adjoint realization of the Schr~dinger operator (1.1)
H = -6 + c sin br + Vex), r
where V is a short range potential of the type introduced by Agmon [1]; i.e., V is a real valued function in L~ (~n) for which there exists an E > 0 so that ;coc
(1.2)
(1 + Ixl)I+E Vex)
defines, as a multiplication map, a compact operator from the Sobolev space into L2.
~2
Schrodinger operators with oscillating long range potentials have been well studied during the past decade [1-11, 14-17]. In a recent interesting paper, Monique Combescure [4] has studied, among other things, a spectral and scattering theory for operators of the form (j,
(1.3)
H = -6
+ c sin br S r
+ Vex).
However, her results are valid only for a linear combination of (j, and S sufficiently large, and do not include the case a = S = 1. In case V is radially symmetric, the spectral and scattering theory for operators of the form (1.3) has been worked out for essentially all values of (j, ~ 0 and 0 < S 2 1 (see [2] and [17]).
107
108
A. DE VINATZ and P. REJTO
In another recent interesting paper, K. Mochizuki and J. Uchiyama [11] have, among other things, considered a spectral theory for operators of the form (1.1). We shall compare our results with theirs after we state our main theorem. In the theorem which follows, and in the remainder of the paper we shall take (1.4)
-11
<
arg z
~ 11,
and (1. 5)
where Ai(z) is the usual Airy function. The constants A and YO are, at least in principle, computable, but we have made no effort to obtain their values. THEOREM 1.1. The interval [AQ,b 2 /4], AO > 0, is in the absolutely continuous spectrum ~ H if
I c I /1 -2v '0 r:;-
(1. 6)
YAO
do ,<1. (1 _ ()'2) '"2
If Al > b 2 /4, then [A 1 ,00) is in the absolutely continuous spectrum of H if b2
( - +1)
46
/o
(1. 7)
k
2
do
<
1,
where <5 = b 2 /4 - AI' The eigenvalues of H in ~ of these intervals is ~ discrete set in the ~ that every compact subinterval contains only ~ finite number ~ eigenvalues (including the mUltiplicity). REMARKS. 1. For fixed c and 6, the inequality (1. 7) is satisfied for b sufficiently large. The number b 2 /4 seems to playa special role since it may be an eigenvalue for H when V = O. 2. In the special case of the operator (1.1) where our results can be compared with those of Mochizuki and Uchiyama [11], we note the following improvements: a. A result on absolute continuity is available in the interval (0,b 2 /4). b. The interval [Al,oo) is independent of the rate of decrease of the short range potential V. Compare [11], pp. 339-340, especially formula (9.12). 3. If E > 1/2 in (1.2), then the location of Al given in Theorem 1.1 is qualitatively the same as given in [11]. 4. If we demand slightly more stringent hypotheses, then a limiting absorption principle is available in all of (0,00). For example, if we take a potential of the form 1jJ(r) sin br + Vex), r
where 1jJ(r) = 0(1),
(1. 8)
(1.9)
~ r '
.
r + 00,
are integrable near infinity,
then [0,00) is in the absolutely continuous spectrum of H. Actually, if we only demand 1jJ(r) = 0(1) instead of (1.8) and set c = lim supl1jJ(r) I, then Theorem 1.1 r+<x>
is true in this situation.
Compare [11] p. 399, formula (9.12).
5. This paper is a continuation of [9] (see also [8]) where we considered
A SCHROEDINGER OPE RA TOR H'ITH AN OSCILLA TING POTENTIAL
109
potentials of the form c sin bra/r B for suitable values of a and B, but without the short range term Vex).
§2
OUTLINE OF PROOF OF THEOREM 1.1
The proof of Theorem 1.1 is relatively long and depends on estimatp.s obtained in [9] for the resolvents of the reduced operator HO
(2.1)
=
-b + c Si~ br
Since the operator (2.1) has a radially symmetric potential, as is well known, this operator is unitarily equivalent to a direct sum of ordinary differential operators defined on the positive real axis R+:
(2.2)
where (taking n
3 for simplicity) j (j + 1) + c sin br d2 ~ + r2 r
(2.3)
Let 1 be a compact interval in JR+ which does not contain an eigenvalue of RO' and let
:R ± (g
(2.4)
) = { z: Re z ('
be rectangles in the complex plane. mer)
(2.5)
=
g,
0 < ± 1m z < a}
Further let 1
+ r,
r E JR+.
A limiting absorption principle in our context means that for every A t E+ which is not an eigenvalue of Ro ' it is possible to find two operators +
-s 2
RO(A): m L
(2.6)
~
s 2
mL ,
s > 1/2,
so that s lim I m- [Ra (A ± iE) - R~(A)]m-sll
(2.7)
E
where Ra(z) is the resolvent of RD. to obtain the inequality
The main step in proving (2.7) for A Egis
sup II m-sR (z)m-s II a zE :R±(g)
(2.8)
= 0,
+a
< "".
In turn this is equivalent to obtaining the estimate (2.9)
where ROj~(z) is the resolvent of R £ Oj the bound in j and z.
The important point is the uniformity of
The resolvent R 'o(z) is an integral operator with a kernel given by OJ x-
(2.10)
ROj~(Z)(~'l1)
~{
f 0 (Of"" (11) W(fo,f",,) fa (11)£",,(0 W(fo,f",,)
~
..::.
11,
11
<
~
.
110
A. DEVINATZ and P. REJTO
The functions fO (~) = fO (j, z) (~) and f (~) solutions to the equation H . u = zu ~ith OJ9, lim
(2.11)
r->-
fa (r) = 0
f",(j,z) (n) are linearly independent
and
f", E L2 (1,"').
a
The denominator in (Z.lO) is the Wronskian of fO and f",. In order to get a uniform estimate such as (2.10) it is, of course, enough to get uniform estimates on the quantities which appear in (2.10). In [9] we did not attack this problem directly, but instead replaced the potential p(j)-z=
j(j +1) + c sin br r2 r- z
by an approximate potential q(j,z) and found linearly independent solutions kO and k", to the equation u" + (z - q(j,z»u = 0,
(2.12)
which satisfied the conditions (2.11). In [9] we constructed functions w, vo and v", on ~+, which we shall not specify here, so that the following lemma holds.
Iko (r) I
<
(Bo + s)w(r)e
Ik", (r) I
<
(B", + s)w(r)e
(Z.13)o,,,,
Moreover, for 11 .:::.
(2.14)
v (r) 0
,
-v (r) co
,
BO
max{A, (411)
B
max{Z1I
'"
'" 2
-!:2 },
A, I}.
~,
IkO (n)k", (0 I IW(kO,k,.)I
<
From these estimates we were able to obtain (2.9) under considerably less stringent hypotheses than in Theorem 1.1. For the purposes of this paper it is necessary to have estimates on the actual resolvent kernel, rather than on the resolvent kernel for an approximate potential. LEMMA Z. 2. Under the hypotheses of Theorem 1.1, for every e: > j 0 ~ that for j ~ j 0 each differential equation
(2.15)
a there exists
~
u" + (z - p(j»u = 0,
e:)w(r)e
(2.16)0
£)w(r)e
(2.16)",
vO (r)
,
-v (r)
'"
.
Further, there exists an M > 0 so that
(2.17) We obtain the estimates (2.16)0 '" by the method of variation of parameters. More specifically we rewrite the equ~tion (2.15) as
(2.18)
u" + [z - q(j,z)]u + [q(j,z) - p(j)]u
o.
111
A SCHROEDINGER OPERATOR h'lTH AN OSCILLATING POTE\lTlAL
For the sake of simplicity of notation write q _ q{j,u), p - p{j). integral equations u{1;)
(2.19) 0,=
I; k (n)k (I;) o =
ko ,= (.;) +
f
+
f
I;
W(ko,koo ) ko (I;)koo (n) W{ko,koo )
If the
[p- q](n)U(ll)dn
[p- q](n)U(n)dll
have solutions, then it is a straightforward computation to show that these solutions are solutions to (2.15). It is not difficult to show that these integral equations have solutions. This does not involve any conditions on the constant c as given in Theorem 1.1. However, in order to get the uniform estimates (2.16)0 and (2.17) it seems to be necessary to invoke the hypotheses of Theorem 1.1. Th~oo estimates of Lemma 2.2 are obtained by using the estimates of Lemma 2.1 in (2.19)0,00' In both of the previous lemmas we have dealt only with ~+(~) in order to simplify the statements. However, corresponding lemmas hold as well for
JI_ (~ ) . Let us continue to work with ~+(~) for simplicity of statements. Let z t ~lt(g) and R(z) the resolvent of H. The second resolvent equation may be written as R(z) - RO (z) ; RO (z) VR(z), or what is the same thing [1- Ro (z)V]R(z) ; Ro (z). Multiplying this last equation on the left and right by m- s , s > 1/2, we get -s -s 2 s -s -s -s -s [I - m RO(z)m m V]m R(z)m ; m RO(z)m .
(2.20)
Let liS set R6(z) ; RO(z) for z t ~(g) and R6(A) the operator given by (2.6) and (2.7) for A t g. Thus from (2.20) we have a limiting absorption principle for R(z) provided [I - m-SRb(z)m-Sm2SV] has a uniformly bounded inverse on ~(g) as a map from H2 ~ H2. Let us set T(z); m-SR1(z)m-sm2sV, z t ~(g). By the hypothesis on V, T(z) is a compact operator from H2 ~ H2. Now, for 1m z # 0, m-SRb(z)m-S is a bounded map from m- s L2 onto a dense set in H2. Thus from (2.20) it follows that the range of 1- T(z) is dense in H2, so that the closure of this range is all of H2. By Fredholm theory the inverse exists as a bounded operator from H 2 to H 2. Thus it remains to show that for A t g , 1- T (A) is invertible. LEMMA 2.3. If the hypotheses of Theorem 1.1 are satisfied and A belongs ~ one the inter~ls of that theor~, then T(A) hilli. @. eigenvalue ilh 1 ~ i f H has ~ eigenvalue at A.
.2..t
The proof of this lemma is too lengthy and technically complicated to even give in outline form. However, we shall give some very broad indications of how it proceeds. As is well known, L2 (:m3 ) is unitarily equivalent to a direct sum of spaces L2 (JR+); i.e.
N. 2 3 L(JR):::
" J L L j;l £;1
where Hj£ ; L2 (JR+) for each j ,L Since Ho has at most one eigenvalue in (0,00) at b 2 /4, and since A i b 2 /4, Rb(A) exists and Hj £ reduces R~(A) to the integral operator R+ .,(A), as we have noted before. 0] "-
For any u E H2 let ~j£ be the component of Vu in Hj £. Then, of course IIVuI1 2 ; II ii. ,112. Let €O be the number given in (1.2) and take 2s ; 1+ €O', €O' ; E/3.
L'J,....,
If
IN
u is an eigenvector for TCA) at the eigenvalue 1, we have -s + -s 2s (2.21) u; m Ro(A)m m Vu.
112
A. DEVINATZ and P. REJTO
At the level of the j, Sl
angular momentum space this gives -s + -s 2 s m ROjSl(A)m m u jSl '
(2.22) ,
JSl
+
Since the kernel (2.10) of ROjSl (z), z t ~+(g), can be extended continuously to 2 ~ (g), and since m S~jSl t L1 ( R+), i t follows that for ~ t R+, -s + -s 2sm ROjSl(A)m m ujSl(f;) (2.23 )
+ m-s(O
OOJfO(Ofoo(1) I; W(fo ,foo)
s m (1)U jSl (1)d1) ,
It is a consequence of abstract considerations (see [12]) that [1- T(A)] implies that
0
o.
(2.24 )
Using this in (2.23) leads to the expression (2.25)
-s+ -s2sm R (A)m m u (I;) OjSl jSl
Using the estimates of Lemma 2.2 in the last formula it can be shown that for every a > 0, there is a Co > 0 and M£, > 0 so that 11m £ ' ull2.. M£,(ollllull + collull).
(2.26)
Also from the equation (2.21) we see that (2.27)
Using (2.26), (2.27) and a "bootstrap" argument we find that for every there is an M > 0 so that o
0
t
lR
(2.28)
This in conjunction with (2.27) shows that A is an eigenvalue of H. Let us now introduce the notation (2.29)
LEMMA 2.4. Suppose the hypotheses of Theorem 1.1 are satisfied and A belongs to one of the intervals of that theorem. g A is ~ eigenvalue of H and u the mrespOriding eigenvector, then for every 0 t R there is ~ Mo ~ that (2.30)
Moreover, T(A) has
II
~
u 112 ,o ::.. Moll ull·
eigenvalue at 1.
The proof of this lemma proceeds very much in the spirit of the proof of the last lemma, using a formula analogous to (2.25), the estimates of Lemma 2.2, and a "bootstrap" argument. The estimate (2.30) and a well known argument due to Agmon [1], shows that the eigenvalues of H form a discrete set in the intervals specified in Theorem 1.1. Further the formula (2.20) now shows that we have a
A SCIIROEDING}-;'R OPLR.-j TOR II'lTH /IN OSClLLAJJNG P071,NiL-lL
113
limiting absorption principle for H in the subintervals of the intervals designated in Theorem 1.1 which do not contain eigenvalues of H. This then will prove Theorem 1.1. More complete details of the proofs will appear elsewhere. REFERENCES [1]
Agmon, S., Spectral properties of Schrodinger operators and scattering theory, Ann. Scuol. Norm. Sup. Pisa 2 (1975) 151-218.
[2]
Ben-Artzi, M. and Devinatz, A., Spectral and scattering theory for the adiabatic oscillator and related potentials, J. Math. Phys. III (1979) 594-607.
[3]
Bourgeois, B., Quantum-mechanical scattering theory for long range oscillating potentials, Thesis, University of Texas at Austin, 1979.
[4]
Combescure, M., Spectral and scattering theory for a class of strongly oscillating potentials, Commun. Math. Phys. 73 (1980) 43-62.
[5J
Combescure, M. and Ginibre, J., Spectral and scattering theory for the Schrodinger operator with strongly oscillating potentials, Ann. Inst. H. Poincare 24 (1976) 17-30.
[6J
Dollard, J. and Friedman, C., Existence of the Moller wave operators for VCr) = Ar- S sin(~ra), Ann. Phys. III (1978) 251-266.
[7J
Devinatz, A., The existence of wave operators for oscillating potentials, J. Math. Phys. 21(9) (1980) 2406-2411.
[8J
Devinatz, A. and Rejto, P., Schrodinger operators with oscillating potentials, in Classical, Semiclassical, and Quantum Mechanical Problems in Mathematics, Chemistry and Physics; K. Gustafson and W. P. Reinhart, eds., Plenum 1980/81 (Proceedings of a meeting in Boulder, Colorado, March 1980).
[9]
_____ , A limiting absorption principle for Schrodinger operators with oscillating potentials, Preprint, 1981.
[lOJ Matveev, V. B., and Skriganov, M. M., Wave operators for the Schrodinger equation with rapidly oscillating potential, Dokl. Akad. Nauk SSSR 202 (1972) 755-758. [11] Mochizuki, K. and Uchiyama, J., Radiation conditions and spectral theory for 2-body Schrtldinger operators with "oscillating" long range potentials I, J. Math. Kyoto Univ. 18-2 (1978) 377-408. [12] Rejto, P., On partly gentle perturbations I, J. Math. Anal. Appl. 17 (1967) 435-462. [13] _____ , Some potential perturbations of the Laplacian, Helv. Phys. Acta 44 (1971) 708-736. [14] Schechter, M., Spectral and scattering theory for elliptic operators of arbitrary order, Comment. Math. Helv. 49 (1974) 84-113. [15] _____ , Wave operators for oscillating potentials, Letters Math. Phys. 2 (1977) 127-132. [16] Skriganov, M. M., Spectrum of the Schrodinger operator with strongly oscillating potentials, Trudy Stek. Math. 125 (1973) 183-195. [17] White, D., Spectral and scattering theory for oscillating central potentials, Thesis, Northwestern University, 1980. lResearch partially supported by NSF Grant MCS 79-02538-AOl. ZResearch partially supported by NSF Grant MCS 78-02l99-AOl.
This page intentionally left blank
This Page Intentionally Left Blank
Spectral Theory of Differential Operators I. W. Knowles and R. T. Lewis (eds.) © North-Holland Publishing Company, 1981
ON CERTAIN REGULAR ORDINARY DIFFERENTIAL EXPRESSIONS AND RELATED DIFFERENTIAL OPERATORS W. N. Everitt Department of Mathematics University of Dundee Dundee DOl 4HN Scotland Dedicated to the memory of Edward Charles Titchmarsh 1899-1963 1.
INTRODUCTION
The purpose of this paper is to study some properties of a second-order linear differential equation of the form M[yJ = AS[yJ
on
[a,bJ
(1. 1 )
where M and S are symmetric (formally self-adjoint) quasi-differential expressions of the second-order and first-order (possibly zero-order) respectively, \ is a complex-valued parameter and [a,bJ is a compact interval of the real line R. Included in (1.1), as a special case, is the generalized Sturm-Liouville equation (1. 2) -(py')' + qy = \wy on [a ,bJ. This last equation is called right-definite if w ~ 0 on [a,bJ, and left-definite if p ~ 0 and q ~ 0 on [a,b]. In either case it is possible to study the equation in the framework of an appropriate Hilbert function-space, and to define differential operators, with discrete spectra, whose eigenvectors may be identified with solutions of the differential equation (1.2) satisfying certain boundary conditions. These ideas are considered here for the equation (1.1) where M and S are taken to be general symmetric differential expressions with complex-valued, Lebesque integrable coefficients. In these circumstances the choice of boundary conditions and a suitable Hilbert space involve a number of interesting questions. We consider in some detail the left-definite problem and then indicate the arpropriate changes in the argument to be made in the right-definite case. The methods used are mainly those of Titchmarsh [22] and Everitt [lOJ. However, it is of some interest to note that the classical methods of Titchmarsh, as in [22, chapter IJ, can be applied only in part to the study of general equations of the form (1.1), or even (1.2); the analytical difficulties of finding asymptotic 115
WN. EVERITT
116
expansions of solutions of these equations, for large values of the spectral parameter A, seem to present problems which require other methods to be adopted. At certain points in the paper we indicate the reason for these difficulties arising, with reference to examples. There has been much recent work on ordinary differential equations of the form (1.1) and the list of references at the end of this paper is in no way complete in this respect. A more comprehensive list of references will be found at the end of the paper by Coddington and de Snoo [5]. See al so the very recent survey paper by Schneider [25]. In section 2 we define the symmetric differential expressions M and Sand consider properties of the resulting quasi-differential equation (1.1); rightdefinite and left-definite problems are defined in section 3, and then considered separately in sections 4 and 5; general remarks are made in section 6. NOTATIONS. Rand C denote the real and complex number fields respectively, with i representing the complex number (0,1); Land AC denote Lebesque integration and absolute continuity; 'loc' denotes a property satisfied on all compact subintervals of an arbitrary interval of R; an expression such as '(x E 1)' is to be read as 'for all elements x in the set I'. 2.
DIFFERENTIAL EQUATIONS AND EXPRESSIONS.
The general symmetric (i.e., formally self-adjoint) differential expression of arbitrary order is given by the Shin-Zettl theory of such expressions; for details see the survey paper by Everitt and Zettl [14], and for the special case of symmetric differential expressions with real-valued coefficients only, see Naimark [16, sections 15 and 16]. Here we consider only first-order and secondorder differential expressions on an arbitrary interval I of the real line R; let the end-points of I be a and b with -00 < a < b ~ 00; in later sections I = [a,b] will be taken as compact. (a)
The differential eXQression M
Let the coefficients p, q and r be defined as follows:
(i)
p:
-T
R and p-1 ,
(i i )
q:
-T
R and q
E
Lloc(I)
(i i i)
r:
-T
C and r
E
Lloc(I);
i.e. , lip,
E
Lloc(I) (2.1)
note that there is no sign restriction on p, at this stage, and that (i) implies that p(x) f for almost all x E I.
°
f: I
Define the quasi-differential coefficients fer] (r C, by
-T
0,1,2) of M, where
117
RFCUL/IR DIFFUU:'NTl/IL FXl'IH,:SSIONS AND RELATED OI'ERATOnS
f[l]
=
p(f' - rf) (2.2)
(p(f' - rf)}' + rp(f' - rf) - qf on I, where the prime' denotes classical differentiation; define the domain D(M) C ACloc(I) of M by
o(M)
(I) for r = 0, l}; loc define the differential expression M: O(M) -)- Ll (I) by, for all f E D(M) oc M[f] = i 2f[2J = - (p(f' - rf))' - rp(f' - rf) + qf on I. =
(f: I
-+
C
I
f[ r] E AC
(2.3)
(2.4)
The general theory in [14J shows that the linear manifold D(M) is dense in Lloc(I). Note that if the coefficient r is null on I then M in (2.4) reduces to the generalized Sturm-Liouville differential expression or the left-hand side of (1.2).
The differential expression M is symmetric in the sense of the following form of Green's formula
s J (g M[f]
f M[g])
(2.5)
a
valid for all compact [a,S] ~ I, and all f, g E D(M); here, as above, the complex conjugate of g, etc.
9 denotes
The differential expression M has a Dirichlet formula /
{P-lf[l] gel] + qf[O] g[D]}
/
a
(p(f' - rf)(g'
rg) + qfg
a
(2.6)
val id as for (2.5). (b)
The differential expression S
Let the coefficients p and w be defined by (i)
p:I->-RandpEACloc(I)
(ii)
w: 1-> Rand WE Lloc(I)
(2.7)
Define the quasi-differential expression S by (2.8)
and for all f E O(S) S[f] = i (pf)' + ipf' + wf note that pf E ACloc(I), since both
p
on
I;
and fare ACloc(I), and that D(M)
(2.9) C
D(S).
The differential expression S is also symmetric with a Green's formula
118
W.N. EVERITT
B
f {g a valid for all compact [a,S]
S[f] - f S[g]}
~
=
g) ISa
(2iof
(2.10)
I and all f, g E D(S).
There is no Dirichlet formula, corresponding to (2.6), for the expression S. There is no loss in generality in taking the coefficient p to be real-valued on I; the expression i(crf)' + i0f' + vf, with cr: I ~ C, is symmetric on I but reduces to (2.9) with 0 = re[cr] and w = v - im[cr']. The differential equation M[y] = AS[y]
(c)
~
I
For A E C consider the linear, second-order differential equqtion M[Y]
=
AS[y]
on
(1.1)
or, equivalently, - (p (y
0
ry))
_
0
_
rp (y
0
_
ry) + qy = A{i (py)' + i py + wy} 0
on
1.
(2.11)
Note that if rand p are null functions on I, then (2.11) reduces to the generalized Sturm-Liouville equation (1.2); also if, additionally, p = w = 1 and q is continuous on I then (2.11) reduces to the Titchmarsh equation studied in [20]. To study the existence and properties of soluti9ns of the general equation (2.11) it is necessary to write the equation in system form. Let Y = [Y1Y2]T be a 2 x 1 column matrix of functions Yl and Y2 defined on I, and consider the first order linear system yo = AY on (2.12 ) where the 2 x 2 matrix A is defi ned on I by
A=
[
c -
. -1 lApp
q - R
-1
P
-r - iAPP
-1
]
(2.13)
with
R=
AW
2 2 -1 + iAp(r - r) + A p P
on
1.
(2.14 )
From (2.1) and (2.7) it follows that R E Lloc(I) and then that the matrix A E Lloc(I); note also that the matrix A is holomorphic on C as a function of the variable
A.
The standard existence theorems for linear differential systems, see Coddington and Levinson [3, chapters 2 and 3], Eastham [9, chapter 1] and particularly Naimark [16, section 16], imply that given any point k E I and two complex numbers a, and a2' there exists a uniquely determined solution Y(x,A), defined for all x E I and all A E C, such that (i ) (i i)
Y (.,A)EAC (I) r l oc y (k,A) = a r
r
(r
1, 2 and A E C)
(r
"
2 and A E C)
(2.15)
119
REGULAR DIFFERENTIAL EXPRESSIONS AND RELATED OPERATORS
(iii)
(r
Yr{x,·) is holomorphic on C
= 1,
2 and x
E
I).
From this existence result and (2.13) it follows that, for all A E C,
i.e.,
.
-1
-1
Yl
(r - lApP
)Yl + P Y2
Y2
p(Yl-rYl)+iAPYl
(2.16)
(q - R)Y l - (r + iAPp-l )Y2' Substituting in this last expression for Y2 from (2.16) gives, after a reduction, on I (2.17)
and
Y2
Now Yl E ACloc(I) and, by hypothesis, p E ACloc(I); hence PYl E ACloc(I) and since Y2 E ACloc(I) it follows that p{Yi - rYl) E ACloc(I); thus the two terms on the left-hand side of (2.17) are separately differentiable (almost everywhere on I) and (2.17) may be re-arranged in the form, writing now Y for Yl' -(p(y' - ry»' - rp(y' - ry) + qy = A{i(py)' + ipy' + wy}
on
I; (2.11)
this yields the equation (2.11) or, equivalently, (l.l). It is noted here that the condition P E ACloc(I), see (2.7)(i), is not essential to the existence of solutions of the system (2.12); for example, if p is measurable and locally bounded on I then the matrix A is Ll (1) which is the oc essential requirement for existence. However, p E ACloc(I) is essential to 'disengaging' the terms p(Yl - rYl) and iAPYl in (2.17) and hence to writing the equation in the form (2.11); furthermore this condition on p is required for the development of the left-definite boundary value problem, associated with the equation (1.1), as given in section 4 below. The initial conditions (2.15){ii) on the components Yl and Y2 of the system (2.12), and the expression (2.16) for Y2' lead to a suitable definition for the quasi-derivatives of a solution y of the second-order scalar equation (2.11); define for all x E I and all A E C the quaSi-derivatives yeO] and y[l] A
y~O](X'A)
y~lJ(x'A)
=
Y (X,A) l p(y' - ry)(x,A) + i\p(x)y(x,A). =
y(X,A)
A
=
(2.18)
The notation indicates that the quasi-derivatives of a solution of the equation (l.l) depend, in general, on the parameter A, and also serves to distinguish the quasi-derivatives of the differential equation (1.1), or equivalently (2.11), from the first two quasi-derivatives of the differential expression M, as given in (2.2); but note that the two sets of quasi-derivatives become identical when p
(x) = 0 (x
E
1).
The existence theorem for a solution of the system (2.12), see (2.15), yields the following existence theorem for the second-order scalar equation(2. 11);
120
W.N. EVERITT
let k E I and a ' a be two complex numbers, then there exists a unique solution O l y(X,A), defined for all x E I and all A EO C, such that
y~r]("A)
(i)
E
ACloc(I)
(r = 0,1 and A E C)
(equivalently y(. ,A), (py)(' ,A) and p(y' - ry)(· ,A) are all ACloc(I) for all A E C)
(i i )
y~r](k,A)
(i i i)
y~rJ(x,' ) is holomorphic on C
a
r
(r = 0,
and A E C)
(r
and x
=
0,
EO
(2.19 )
I).
The requirement (2.19)(ii) is called a set of initial conditions at the point k for a solution of the equation (2.11). The generalized Wronskian, see [16, section 16.3J of two solutions y(. ,A) and z(· ,A) of (2.11) is defined to be (2.20) for all x EO I and all A E C. differentiate (2.20))
A calculation shows that (details are omitted but
f
W(y,Z)(X,A) = W(y,z)(k,A)exp[2i
x {im[r] - APp-l}]
(2.21)
k
for x, k E I and A E C. If y and z satisfy initial conditions of the form (2.19) (ii), at the same or different points, then W(y,z)(x,·) is a holomorphic function on C, for all x EO I. The standard relationship between the ~lronskian of two solutions vanishing at one (equivalently all) point of I, and tne linear dependence of solutions of the differential equation (2.11), continues to hold; see [16, section 16.3, theorem 3]. The introduction of the (in general) complex-valued coefficient r into the symmetric differential expression M, see (2.4), has an advantage in addition to that of generality; it allows of translating the A parameter in (2.11) along the real axis of C and yet retaining the same form of the differential equation. If A = J.l + , where J.l E C and, is a real number, i.e., the translation, then the differential equation - (p (Z r ,z))' - r, p (z' - r , z) + q, z = J.l {i (p z ) + i p z' + wz} 0 n I, (2.22) I
-
I
where the new coefficients on the left-hand side are given on I by .
r,=r+l,pp q,
-1
q + ,w - 2,im[r] _ ,2p2p-l,
(2.23)
has solutions given by z(x,J.l) = y(x, for all x
E
I and all J.l
E
C.
J.l
+ ,)
This result follows on substitution in (2.23) from
121
REGULAR DIFFHRHN'IlAL CXI'RllSS[ONS AND REL4Tl'D OPERATORS
from (2.24) and simplification of the terms. There is a differential equation of similar form which can be associated with (1.1) or (2.11); this equation is obtained by taking complex conjugates of (2.11) and then replacing I by),; this gives the associated differential equation -(p(z' - rz))' - rp(z' - rz) + qz
=
),{-i (pz)' - ipz' + wz}
on
(2.24 )
or, say, N[zJ = )'T[zJ
on
with Nand T symmetric differential expressions of the second-order and firstorder respectively. Note that (2.24) is obtained from (2.11) by replacing r with rand p with -po If Y(X,A) (x E I, A E C) is a solution of (2.11) then Z(X,A) = y(x,I) is a solution of (2.24). This leads to an identity concerning solutions of the differential equation (2.11) which generalizes a property of certain solutions of the Sturm-Liouville equation (1.2), represented by the result ~(x,I) = ¢(x,),) (x E I, A E C) for a solution ¢ of (1.2) taking real initial conditions at a point k E I; see, for example, [22, (1.6.1) and line 5 of page l2J. The corresponding result for the differential equation (2.11) is stated in the following: Lemma. Let aO' al E R and let ¢ be the solution of the differential equation (2.11) which takes the following real initial conditions at a point k E I
¢~OJ(k,)) = ¢(k,A) = aO 'P
[1 A
J (k,A)
p(¢' - r¢) (k,A) + iAP(k)¢(k,A) cjJ(X,A) exp[-
;j;( x, I)
f
a ; then for all x l
=
x
(2.25)
_
{r - r - 2i),pp
-1
E
I and all ),
}].
E
C
(2.26)
k
Proof. Clearly ~(. ,I) is a solution of (2.24). and then complex conjugates taken we obtain -;P(k,I) p(~'
-
~)(k,I)
-
= a
If A is replaced by I in (2.25)
O
i),p(k)~(k,I)
= al ;
the left-hand sides of these last results are the quasi-derivatives of a solution of the associated equation (2.24). x
E
(2.27 ) ~(.
,I) as
On the other hand if ¢ is defined by the right-hand side of (2.26), for all I and A E C, then differentiation yields p(¢' - r¢)(x,A) - iAP(x)rp(x,A) =
(p(cjJ' - rq,)(x,A) + iAP(x)q,(x,A))exp[···J
(2.28)
a second differentiation, together with the elimination of cjJ as a solution of
122
W.N. EVERITT
(2.11), gives ¢ as a solution of (2.25). From the definition of ¢ and (2.28) it follows that ~ satisfies the same initial conditions, as a solution of (2.25), as given by (2.27) for ;(. ,i). From the uniqueness of solutions of (2.25) satisfying the same initial conditions it now follows that (x E 1, A E C)
"¢(x,i) = rJl(x,A)
and this completes the proof of the lemma. If r is real-valued and p is the null function on I, then the differential equation (2.11) has all real-valued coefficients and (2.26) reduces to the result ;(. ,i) = q,(. ,A) previously mentioned. Note that the real initial conditions are essential to this last result and to (2.26). The transformation theory of the quasi-differential equation (2.11) is considered in detail in Everitt [12J, where the transformations given are related to the left-definite and right-definite boundary value problems to be introduced in the next section. (d)
The equation S[wJ = 0
Finally in this section we look at the possibility of solving the equation, not necessarily a differential equation, S[wJ = 0
on
(2.29)
where S is the first-order symmetric differential expression defined in (2.9). The condition (2.7)(i) on the coefficient p does not exclude the possibility of vanishing at a point of I, or even on a sub-interval of I; in general then (2.29) is not a first-order differential equation on I. For application in subsequent sections we need only consider (2.29) in the case when I is compact, i.e., it is now assumed that
= [a,bJ;
(2.30)
the coefficient p then satisfies the condition p E AC[a,bJ.
(2.31 )
At this stage it is convenient to exclude the following case for the coefficients p and w of the differential expression S p(x) = 0 (x
E
[a,b])
and
w(x) = 0 (almost all x
E
[a,b])
(2.32)
although either one or the other may hold separately; this is to exclude the case when Sew] = 0 for all WE AC[a,b], otherwise· the differential equation (1.1) reduces to M[yJ = 0 on [a,b] and is consequently independent of A. The coefficient mutually exclusive,
p
may be classified as follows, where the three cases are
RFoGULAR DIFHiRIiNTr'lL liXPRJ:SSJONS _1,\'D REL4 now OPERATORS
(i)
pix)
or
(i i)
r(x)
or
(i i i)
either
0 (or
>
0)
<
0
(x
EO [a,b])
(x
EO [a,b])
123
there is a non-empty, at most countable set P of open subintervals of [a,b], say {( a r , Sr) : r E Pl, satisfying a ~ ar
<
pix)
0
>
Sr ~ ar+l (or
<
(x
0)
(r EO P)
Sr+l ~ b
<
r EO p)
E (ar'Sr)
and
p(x) = 0 Consider now the solution of (2.29) under the restriction ~ E AC[a,b]; define the linear manifold Gp,w by (note that the null function on [a,b] is always a member of G) G p,w
{~ E
AC [a , b]: S[ 'v ] ( x)
= 0 ( a 1mo s tall
x EO [a, b] ) }.
(2.33)
Case (i) Here (2.29) is a regular, first-order differential equation on [a,b] with solution ~(x) =
K{p(x)
for any K EO C; in this case G
p,w
-1/2
1
x
-1
eXP[2 J rip wl]
(x EO [a,b]) (2.34) a is exactly the linear manifold of these solu-
tions. Case (ii)
Here (2.29) requires of w(x)~(x)
=
~
that
(almost all x
0
E
(2.35)
[a ,b]);
in this case w is not null on [a,b], see (2.32), so that (2.35) implies some restriction on ~, but note that in any set of positive measure where w is null then ~ is unrestricted, but always subject to ~ EO AC[a,bJ. Case (iii)- Here (2.29) is a regular first-order differential equation in the open intervals {(ar'Sr): rEO PJ with solutions of the form (2.34); however since p(a ) = 0 or pis ) = 0 the only choice for K which keeps the solution in r r AC[a r 'Sr J is K = 0, i.e., if ~ E Gp,W then 'V(X)
On the closed set F
C
[a,b] the solution w(xl~(x)
0
0 ~
(x
E
[ar'Sr]
r EO
Pl.
(2.36)
must satisfy additionally
(almost all x E F)
and again this may impose further restrictions on the function
(2.35) ~.
124
3.
W.N. EVERITT
RIGHT-DEFINITE AND LEFT-DEFINITE BOUNDARY VALUE PROBLEMS It is now assumed for the remaining sections of the paper that the interval
I is compact, i.e., I '" [a ,b]. (a)
Symmetric boundary conditions.
A symmetric boundary value problem for the differential equation
M[Y] '" AS[y]
on
[a,b]
(3.1)
is determined by imposing appropriate homogeneous boundary conditions to be satisfied by the solutions of (3.1). Here these boundary conditions are applied at the end-points a and b of the interval concerned; this compares with the classical Sturm-Liouville problem as conSidered in [21, chapter I] for which the differential equation is -y" + qy = Ay on [a,b] (3.1a) with q real-valued and continuous on [a,b]. The general theory of symmetric boundary conditions for differential equations of the form (3.1) has been developed by Pleijel in a number of papers; see the results [18J and [19]. In this paper symmetric boundary conditions are derived from the theory of quasi-differential equations; the quasi-derivatives of (3.1) are required to satisfy certain separated boundary conditions at the endpOints a and b. One of the advantages of this particular type of boundary condition is that they determine symmetric boundary value problems for both the leftdefinite and right-definite cases, to be defined below, of the differential equation (3.1). They also reduce to the classical Sturm-Liouville boundary conditions when the special case (3.1a) of (3.1) is considered. Let y, 0 E [- ~1f' ~1f]; then separated symmetric boundary conditions for a solution y of the differential equation (3.1), at the end-points a and b, are given by yeo] (a) cos \
y
_ y[1] (a) si n
y[O] (b) cos
<5
+
\
y
o
(b) sin Ii
0;
A
yClJ A
here the quasi-derivatives of yare defined in (2.18). y these boundary conditions take the equivalent form yea) cos
y -
iAp(a)y(a) sin
y -
(3.2)
In terms of the solution
ply' - ry)(a) sin
y
0
with a similar form at the end-point b. The parameters y and 0 in (3.2) are introduced to give a convenient form to 1Y real boundary condition of the form
yeo] o \
A
(a) +
A y[l] (a) 1
A
0;
125
RliGU[,AR DIFFERliNTIAI. EXPRhSSIONS AND RELITHD OPLR,i'I'ORS
i.e., AO and Al are real and not both zero. It can be shown that for a boundary condition of this form to be symmetric for the equation (3.1), it is essential for Ao and Al to be real so that (3.2) represents the most general form of separated boundary conditions, involving the quasi-derivatives, for the equation (3.1) . The boundary conditions (3.2) reduce to the classical boundary conditions for the equation (3.1a), i.e., from [21, (1.6.3)J y(a) cos
u. +
y' (a) sin a
0
y(b) cos
8 +
y' (a) sin
0
B
(3.2a)
if a = -y and B = 6. The reason for choosing a different sign in the conditions (3.2) at a and b is for convenience in the left-definite problem considered below. It should be noted that, in general, the boundary conditions (3.2) explicitly involve the spectral parameter A.
For a general study of separated symmetric boundary conditions see Pleijel [19J. Non-separated boundary conditions, i.e., those involving the quasiderivatives at a and b together, can also be considered. In this case additional conditions are required for symmetry, but these do allow for the introduction of complex-valued boundary conditions not possible in the separated case. For some details in the Sturm-Liouville case see [22, section 1.14J and [3, chapter 11, problem 1]. (b)
Boundary value problems.
A boundary value problem is now determined by the differential equation (3.1) and, for fixed y and 6, the boundary conditions (3.2). A solution to this problem is a pair (A,
sin
y
(a,A)
cos
xf O]
- sin
(b, \)
cos
A
y
(\ E
C)
(\ E
C).
(3.3) (b,\)
It is clear that ¢, respectively x' satisfies the boundary condition at a, respectively b, for all \ E C. If for some A it can be determined that ijJ and x are linearly dependent solutions of (3.1), say x = k¢ with k E C but k f 0, then
WNE~RnT
126
~ (and also x) is a non-trivial solution of the equation (3.1) satisfying both boundary conditions (3.2). This remark leads to
Lemma 3(b). by
W(x,~)(X,A)
Let the generalized Wronskian W(x,¢)(· ,A) be defined for all A E C
x~O](x'A)~~l](x'A)
[1]
[0]
(x E [a,b]); (3.4) - x A (X,A)¢A (X,A) then W(x,~)(x,·) is an integral (entire) function on C for all x E [a,b]; A E C is an eigenvalue of the problem (3.1 and 2) if and only if for some ~E[a,b] and then for all ~ E [a,b], =
All the eigenvalues of the problem (3.1 and 2) are simple. Proof. See the remarks in section 2, in particular (2.21), for the properties of the Wronskian W. If W(x,¢)("\) is null on [a,b] then x and ¢ are linearly dependent and, as above, A is an eigenvalue. If A is an eigenvalue, with eigenfunction W, and yet W(x,¢)(· ,A) does not vanish on [a,b] then w = Ax + B¢; however a standard argument then shows that A = B = 0 which is a contradiction; thus W(x,¢)(· ,A) is null on [a,b]. If A is an eigenvalue with eigenfunction wthen any solution 8 of the differential equation (3.1), which is linearly independent of W, cannot satisfy the same boundary condition at a (or at b); thus ~ is unique, up to linear independence, and the eigenvalue A is simple. This result is essentially a consequence of adopting the separated boundary conditions (3.2) for the boundary value problem. This completes the proof. Since W(x,¢)(a,·) is an integral function its zeros are either discrete with at most one limit-point at infinity, or W(x,¢)(a,·) is identically zero on C. Little more can be said in general, although it will be shown below that in both the left-definite and right-definite cases W(x,¢)(a,·) is not identically zero but does have an infinity of real zeros and no strictly complex zeros. It does not seem to be known in the indefinite cases of the boundary value problem consisting of the differential equation (3.1) with separated boundary conditions (3.2), if it is possible for W(x,~)(a,·) to be identically zero or to have only a finite number of zeros. If non-separated boundary conditions are introduced then W(x,¢)(a,·) can be identically zero; see [3, chapter 12, section
1]. To make further progress in the study of the boundary value problem (3.1) and (3.2) it seems essential either to embed the problem in a function space setting or to employ classical and asymptotic analytic methods; the first method is
127
REGULAR DIFFEREN'11AL ':XI'RLSS[ONS AND RhLATHD OPERATORS
exemplified in Naimark [16, chapters V and VI], and the second method in Titchmarsh [22, chapters I and II]; the account in Coddington and Levinson [3, chapter 7, and problems 8 and 9 of that chapter] uses methods in both of these areas. All these accounts are concerned with right-definite problems; for operator theoretic results in right- and left-definite cases see [4J, [17J, [18J, and [19J. For a general study of right-definite and left-definite separated boundary value problems, see the survey paper by Pleijel [19J. In this paper the methods of Titchmarsh [22, chapter IIJ are employed, with some use of operator theoretic methods. It should be noted that the classical methods of Titchmarsh [22, chapter IJ lean heavily on the asymptotic expansion of solutions of the differential equation (3.1a) for large values of the parameter A; however, few, if any, results of this kind are known for other forms of the general equation (3.1); this remark even applies to the generalized Sturm-Liouville equation (l .2) when the coefficients are only locally integrable and, say, the Liouville transformation, see [22, section 1.14] cannot be applied; for a survey of transformation results see Everitt [12]. (c)
Definite differential expressions.
Let P represent a general symmetric quasi-differential expression, as defined in Everitt and Zettl [14J, on a compact interval [a,bJ with domain D{P) C AC[a,b]. Then P is said to be non-negative definite on a linear manifold 0 (P) c D{P) if b b a J f P[f] = J P[f] f ~ 0 (f E Do{P») (3.5) a a and the left-hand side is not zero for all f E Do{P). The domain Do{P) can often be determined by restricting the elements of D{P) to satisfy certain separated boundary conditions at the end-points a and b. As an example if P[fJ = -f" on [a,b] with D{P) = {f: [a,bJ + C I f and f' EAC[a,bJ}, let Do{P) = {f E D{P) I f{a) = f{b) = OJ. In the case of the general second-order symmetric differential expression M defined by (2.1 to 4), let Do{M) be determined by imposing separated symmetric boundary conditions of the form (3.2), at the end-points a and b, on the quasiderivatives of the elements of D(M), i.e., f E Do (M) if f E D{M) and frO] (a) cos y - f[l] (a) sin y
0
frO] (b) cos
O.
t5
+ f[l] (b) sin
Then the Dirichlet formula (2.6) yields
(3.6)
W.N. F.VERITT
128
b
J a
f
b
J M[f] f
M[f]
a cot6If[O](b) 12 + coty\f[O](a) 12 +
b
J
{p- l lf[l J i2 + qif[O]1 2}
(3.7)
a where the integrated terms on the right-hand side are to be omitted if 6 0 and/ or y = 0 respectively. From this result it follows that M is non-negative definite on Do(M) if, additionally the following conditions are imposed
p(x) y E
>
[0,
and
0
1
2 nJ
q(x)
~ 0
6 E [0,
(almost all x
E
[a,bJ)
1
2 nJ.
(3.8)
Note that no additional condition has to be placed on the coefficient r of M. In the case of the symmetric differential expression S, of the first (or zero) order, given by (2.9), there is no Dirichlet formula and it may be seen that S can only be made non-negative definite on [a,b] if p(x)
w(x)
~ 0
=0
(x E [a,bJ) b
(almost all x E [a,bJ)
and
J
w(x)dx
>
0;
(3.9)
a
in this case Do(S) can be taken to be D(S) = AC[a,bJ (or an even larger set), i.e. no boundary conditions are requi red. Thus S must be reduced, necessari ly, to a differential expression of zero order for the non-negative definite property to hold. It is worth remarking that it is only symmetric differential expressions of even order (including zero order) which can be non-negative definite; see the result in Dunford and Schwartz [8, chapter XIII, section 7.29, page 1457, lemma 29]. Such non-negative differential expressions can be utilized to define a Hilbert function space which leads to a suitable framework in which to consider properties of the general boundary value problem represented by (3.1 and 2) above. Additional information concerning the number and position of zeros of W(x,~)(a,·), i.e., the eigenvalues of the problem, can then be obtained, furthermore the eigenfunctions can be considered as forming a normal, orthogonal set in this Hilbert function space, which set, however, mayor may not be complete in the space. The boundary value problem (3.1 and 2) is called left-definite or rightdefinite according to whether the symmetric differential expression M or S respectively, is non-negative definite on [a,b].
129
R/;CULAR LJllTERliNn.IL EXPRESS[ONS AND [U'L.1TED OPER:1TORS
(d)
Left-definite and right-definite boundary value problems. The following definitions are now made for the general boundary value
probl em (3.1) and (3.2): Left-definite problem The coefficients p, q, r, p and w of the differential equation (3.1) satisfy the basic conditions (2.1) and (2.7) with I = [a,b]; additionally p. q, p and w satisfy p(x)
>
0
and
q(x)
>
0
(almost all x
E
[a,bJ)
(3.10)
b
(3.11 ) J {\p(x)1 + \w(x)\}dx > 0 a and the boundary condition parameters y, 6 of (3.2) are restricted to satisfy Y E [0,
z1 TIJ
Ii
1 E [0, ZTI].
(3.12 )
Right-definite problem The coefficients p, q, r, p and w of (3.1) satisfy the basic conditions (2.1) and (2.7) with I = [a,b]; additionally p and w satisfy p(x)
(x
[a,b]) (3.13 ) b (3.14 ) w(x) 2:. 0 (x E [a,b]) J w(x)dx > 0 a and the boundary condition parameters y, 6 of (3.2) satisfy, without loss of general ity, y E
1
0
E
1
[-ZTI, ZTI]
(3.15 )
Remarks In the left-definite case there is no restriction on the sign of p or w on [a,b], but the condition (3.11) excludes the possibility that both p and ware null on [a,b]; there is no additional restriction on the complex-valued coefficient r; from (3.10) and the condition p-l E L(a,b) it follows that p(x)
>
0
(almost all x
E
[a,b]).
(3.10) I
In the right-definite case it is essential to take p to be null on [a,b] but there is no sign restriction on either of the coefficients p and q; w is nonnegative but not null on [a,b]. Note that w may vanish on a set of positive measure in [a,b] which relaxes the normal condition on w, i.e., w(x) > 0 (almost all x E [a,bJ), but this implies that the usual method of determining a differential operator through use of the expression W-1M[.], see [14, section 6], is not appropriate; it will be shown below how this difficulty can be overcome.
130
(e)
W.N. EVERITT
Hilbert function spaces.
For both these boundary value problems it is essential to define an appropriate Hilbert function space, in terms of one or more of the coefficients of the differential equation, which allows of the possibility of representing the solution of the problem in operator theoretic terms. In certain cases the boundary value problem can be characterized by means of a uniquely determined unbounded self-adjoint operator in this function space; in these cases the eigenvalues and eigenfunctions of the boundary value problem are equivalent to the eigenvalues and eigenvectors of the operator; these will be called 'self-adjoint' cases of the boundary value problem. In other cases such a characterization is not possible and these will be referred to as 'symmetric' cases in general. These spaces are determined as follows. Left-definite problem For this problem the basic conditions (2.1) and (2.7) hold, together with the specific left-definite conditions (3.10), (3.11) and (3.12); in particular p ~ 0 1 and q .:: 0 on [a,bJ and y, <5 E [0, Z1T]' Define the inner-product space Hy, u,(a,b) = Hy, u,as the following collection of complex-valued functions on the compact interval [a,bJ (i) fEAC[a,bJ (ii) pl/2(f' -rf)EL 2 (a,b) Hy, 8 = {f:[a,bJ-+C I (iii) f(a)
=
0 if
y =
D,
f(b)
=
0 if
8 = O}
(3.16)
with inner-product b
(f,g) y, <5
J
{p(f' - rf)(g'
rg) + qf g} +
a
cot
y'
f(a)g(a) + cot o' f(b)g(b)
(3.17)
where the cot y (cot 8) term is to be omitted if y = 0 (8 = 0). (Note that the integral in (3.17) exists from (3.16) and the condition q E L(a,b).) We remark that this space is chosen to take into account the specific boundary conditions (3.2) of the boundary value problem; this follows the procedure adopted by Everitt [10, section 2J, Atkinson, Everitt and Ong [2J, Daho and Langer [6J and [7J. For comments on this choice of function space, which is essential if the symmetric boundary conditions (3.2) are adopted, see Pleijel [18, section 8]. Clearly, under the given conditions, H is a quasi-inner-product space, see y,6 Akhiezer and Glazman [1, section 3J; to determine if it is an inner-product space it is necessary to identify those elements f of Hy, 6 which satisfy b
Suppose this is the case, then J
(f,f)
8=0. plf' - rfl 2
(3.18)
y,
0 and, from (3.14), this implies
a
f' - rf
=
0 (almost everywhere on [a,b]), i.e., for some K E C
131
Ri":CULAR DIFFERENTJ/IL [iXPRESSIONS AND RELATED OPERATORS
x
fix)
K exp[J a
=
r(t)dt]
(x E
[a,b])
(3.19 )
t
t
If either Y f TI or 6 f TI then the other terms on the right-hand side of (3.17), or the definition (3.16) if Y = 0 or 0 = 0, imply that f(a) = 0 or fib) = 0, i.e., K = 0; thus fix) = 0 (x E [a,b]). If y = 21 TI and a = 21 TI and q( ~ 0) is not null on [a,b] then the integral term in (3.17) implies again that K = 0 in (3.19), i.e., fix) = 0 (x E [a,bJ).
t
Finally if y = TI, 0 = ~ TI and q(x) f of the form (3.19) satisfies (3.18).
(almost all x
= 0
Thus there are two cases to consider: (1) if at least one of 1 1 either y f 2 TI or a f 2
E
[a,bJ), then any
(3.20)
TI
b
J q(x)dx
or
>
(3.21 )
0
a
holds then (.,.) cis an inner-product on H ,as a collection of absolutely cony, y,6 tinuous functions on [a,bJ, with the null element the null function on [a,bJ, i.e. f = 0 in Hy,o if and only if fix) = 0 (x E [a,bJ); (2)
if y
=
cS
1
=2
b
TI
and
J q(x)dx
=
0
(3.22)
a
then (. ,. L. J. is a quasi-inner-product on HJ• J. ; as in [1, section 3J this may "21T ,'z1T "z1T, "z1T be considered as an inner-product on the collection of equivalence classes of functions generated by the null class consisting of all f given by (3.19) for all K E C.
Classical arguments show that in case (1) above Hy,us is complete, [1, section 4J, in the norm derived from the inner-product (. ,.) y,us' i.e., Hy, us is a Hilbert space of point-wise, absolutely continuous functions on [a,bJ. Details of the completion argument are omitted (but see, for example, Coddington and de 5noo [5, p. 158J) except to remark that completion is first proved when r is null on [a,b], and then extended to the general case by means of the unitary (isometric) transx fix) exp[-J r(t)dtJ (x E [a,b]); see Everitt [12, sections formation (Vf)(x) a 4.1 and 5.1]' Similar arguments show that in case (2) HJ J , ' as a collection of equiva;.zlT '~21T lence classes, is complete in the norm derived from the inner-product, and, in this sense, is a Hilbert space. It will be shown below that case (2) is a pathological exception for the general boundary value problem (3.1) and (3.2).
w.N. EVERiTT
132
If from the definition (2.3) of the domain OeM) the domain O,(M) is defined y,u by
oy,u,(M) =
{f
E
OeM) I f(a) =
°
if 'I
= 0, feb) =
°
if 6
= O}
1 then the following relationships may be seen to hold, for all y, 6 E [0, 2 TI],
oy, u,(M) H
'1,6
Hy,o,c AC[a,b] = O(S).
C
(3.23)
There is a need, in general, in case (1) above to reduce the Hilbert space (a,b) in order to take into account the linear manifold G as defined by
r,w
°
(2.33), i.e., the manifold of solutions of Sew] = on [a,b] as discussed in detail in section 2(d) above. This reduction means that, in general, the appropriate Hilbert space for the left-definite boundary value problem (3.1) and (3.2) depends not only on the coefficients p, q and r of M, from the definition (3.16) and (3.17), but also on the coefficients p and W of S. The method used here follows that adopted by Everitt [10, section 2]; see also Oaho and Langer [6]. Suppose case (1) above to hold, i.e., (3.20) and (3.21), and recall the definition (2.33) of G w; define the linear manifold G ,c G by
"v -
p,
Gy,6
{f
=
E
Gp, wi f
E
p,w
Hy,v,L
(3.24)
-
Define the sub-space H,,0,of H'1,6 as the orthogonal complement, see [1, section 7], of G s in H s ' i.e., y~1.)
y~u
H'1,6 We note that H
'1,6
H '1,6
e
{f
Hy,o I (f,g)y,O
E
G y,6
is the largest sub-space of H
,,6
f
E
Hand S[f] = '1,6
=
°
(gEG
,)}.
'1,0
(3.25)
such that
° if and only
if f = 0.
(3.26)
It will be shown in the next section that, under certain additional conditions, the Hilbert space Hy,o,is appropriate for the definition of a self-adjoint operator to represent the left-definite boundary value problem. Right-definite problem For this problem the basic conditions (2.1) and (2.7) hold, together with the specific right-definite conditions (3.13), (3.14) and (3.15); in particular p = andw2-0on [a,b], and"
1
°
1
6E[-2TI'2TI].
In this case the appropriate Hilbert function space is L~(a,b), i.e., the collection of all complex-valued, Lebesque measurable functions f on [a,b] such that
f a
with quasi-inner-product
b
w(x)lf(x)2 Idx
<
~
(3.27)
RFGL'L.4R DIFFEI!ENTL 1L /,'XI'I
133
b
J w(x)f(x)g(x)dx.
(3.28)
a The null element of this space is the linear manifold of all such functions satisfying b
J w(x) 1f (x) 12 dx
= 0; a if w(x) > 0 (almost all x E [a,bJ) then this null element is the set of all f which are zero almost everywhere on [a,bJ; otherwise the null element will contain
certain non-null functions. The elements of the space L~(a,b) are equivalence classes of functions generated by the null element in the usual manner; see [1, section 3J. With this interpretation (3.28) defines an inner-product for L2 (a,b), and the w usual classical arguments show that this space is complete in the norm derived from the inner-product. In this right-definite case there is no essential requirement to reduce the Hilbert space L2 (a,b) by the linear manifold G ,as in the left-definite case; w o,w in fact it may be seen from case (ii) of section 2(d) that G is the above de2 termined null element of L (a,b). However, if it is desiredOt~ work in a space of w point-wise defined functions as elements, rather than equivalence classes as elements, then L~(a,b) can be reduced to a sub-space L~(a,b) by the definition, using the methcx:l of orthogonal complement, -2
Lw(a,b)
2
=
Lw(a,b)
e Go,w
The null element in L~(a,b) is the null function on [a,bJ. It will be shown in section 5 below that every right-definite boundary value problem can be represented by a self-adjoint operator in the associated Hilbert 2 (a,b) or, equivalently, L2(a,b). function space Lw w . (f)
Translation of the spectral parameter A
In section 2 it is shown that it is always possible to translate the A parameter in the equation (2.11), i.e., M[yJ = AS[yJ on [a,bJ, along the real axis of the complex plane and yet retain the symmetric form of the equation; see (2.22) and (2.23). Left-definite case real p (> q, ~ q, .::..
Suppose that the equation (2.11) is in the left-definite case and consider a translation A = v + T; in the translated equation (2.22) the coefficient 0) is invariant, but the coefficient q is changed to q given in (2.23), i.e. q, + ,w - 2, im[rJ - ,2p2p-l and q T mayor may not sati~fY the condition 0 on [a,b]' Thus translation mayor may not leave invariant the left-definite
134
W.N. EVERITT
property of the equation. It can happen that a left-definite problem is not left-definite under any real translation T f 0; as an example consider the left-definite equation -y" = ii-y'
i.e., p = 1, q rea 1 T f o.
r =w
on
[a,b],
0, P = ton [a,b]; then qT = -
t T2
<
0 on [a,b] for all
Right-definite case All right-definite equations (2.11) can be translated along the real axis and retain the right-definite form for all translations; the appropriate translated form is given by (2.22) and (2.23), with P = 0 on [a,b]. 4.
THE LEFT-DEFINITE CASE
This section is devoted to a study of the differential equation and boundary conditions under the left-definite requirements on the coefficients. There are a number of results which can be obtained for the general symmetric boundary value problem; these are given in sections 4.1 to 4.8. In section 4.9 the self-adjoint case is considered; in section 4.10 the main results for the symmetric non-selfadjoint case are given. Finally in section 4.11 a number of examples are 9iven which indicate that the results obtained cannot be improved upon in general. 4.1.
The left-definite boundary value problem.
For convenience the symmetric boundary value problem is restated; the differential equation is M[Y] AS[y] on [a,b] (4.1) or, equivalently, -(p(y' - ry))'
rp(y' - ry) + qy
A{i(py)' + ip'y + wy} on [a,b],
(4.1 a)
with boundary conditions y[O](a) cos
y
=0
y[O](b) cos 6 + y[l](b) sin 6
0
A
A
y -
y[l](a) sin A A
(4.2)
or, equivalently, y(a) cos y (b) co s
y -
(p(y'
ry) + iAPy)(a) sin
(p (y'
ry) + iAPY) (b) sin
y
0 O.
(4. 2a)
The coefficients p, q, r, P and w satisfy the basic conditions (2.1) and (2.7); the left-definite conditions (3.10), (3.11) and (3.12) hold; apart from sections 4.2 and one example in section 4.11, the conditions of case (1) in section 3(e), i.e., (3.20) or (3.21), hold throughout the section 4.
RFGULAR DIFf'ERmVTIAL r;XPRESS[ONS AND RELATFD OPERATORS
4.2.
135
The pathological case
This is case (2) in section 3(e) for which there are ties; the appropriate Hilbert space H, is singular in '"'z1T, "ziT with all the other cases; it is the only case for which 0 zeros of the generalized Wronskian W(x,~)(a,·) may not be I
a number of difficulnature in comparison is an eigenvalue; the simple.
Lemma 4.2. The number 0 is an eigenval ue of the left-definite boundary val ue problem (4.1) and (4.2) i f and only if condition (3.22) holds, i.e., 1
y = 6 = Z
and
rr
q(x)
=
0
(almost all x
E
[a ,bJ).
(4.3)
Proof. Suppose 0 is an eigenvalue; then there is a non-trivial solution ~ of M[yJ = 0 on [a,bJ satisfying the boundary conditions. From the Dirichlet formula (2.6)
since A
0 and b
M[~J =
J {p[~'
0 on [a,b], i.e.,
- r~[2 + q[~[2} + cot o· [~(b) [2 + cot y. [~(a) [2 = 0
(4.4)
a
from the boundary conditions (4.2), with adjustment if y = 0 or 6 = O. Thus, from (3.10) and (3.12), it follows that 1~'(X) r(x)~(x) (x E [a,bJ), i.e., ~(x)
K exp[
f
x
(4.5) (x E [a,bJ) r(t)dtJ a for some K E C with K t O. This implies, from (4.4), that q(x) = 0 (almost all x E [a,bJ), and that y = ~rr (6 = ~rr ) or ~(a) = 0 (~(b) = 0) of which only the former case is possible. Thus (4.3) is satisfied. =
If (4.3) holds then it follows at once that ~ defined by (4.4) is a nono trivial solution to the boundary value problem for A = o. It will be shown in section 4.11 below that the example -y" =
iAy' on [0, 2rrJ
1
y=6=2
11 ,
which satisfies the condition (4.3), yields an orthonormal set of eigenfunctions which is not complete in
H~11,~11.
Thus in general, a left-definite problem in case (2) of section 3(e), i.e., when (4.3) holds, cannot be represented by a self-adjoint operator in the associated Hilbert space H, of equivalence classes. L
YzTI ,/zTI
For the remainder of this section case (2) of section 3(e) is excluded and case (1) holds, i.e., (3.20) or (3.21) is satisfied.
136
4.3.
WN. EVHRITT
Eigenfunctions The following lemma is required.
Lemma 4.3. Let the left-definite conditions hold, including (3.20) or (3.21); let A ( f 0) be an eigenvalue of the problem with (non-trivial) eigenfunction ~; then (~,ljJ)
,
y,v
(4.6)
O.
>
Proof. Suppose to the contrary, i. e., (ljJ, ~J) y,o, 0; then as in the proof of 1emma 4.2 above x (x E [a,bJ) ljJ(x) = K exp[ f r(t)dtJ a with K f 0; it then follows from (4.6) that q 0 on [a,bJ. From (3.21) either y E [0, ~1T) or <5 E [0, ~1T); if y = 0 or <5 = 0 then ljJ(a) = 0 or ljJ(b) = 0 which gives K = 0; if y E (0, t1T) or <5 E (0, ~1T ) then cot Y'lljJ(a) 12 = 0 or 0 cot <5·lljJ(b) 12 = 0 and again K = O. Thus (4.6) must hold. 4.4.
Eigenvalues
Lemma 4.4. All the eigenvalues of the left-definite problem (4.1) and (4.2) are real and simple; the Wronskian W(x,~)(a,·) (see lemma 3(b)) is not identically zero; the zeros of W(x,~)(a,·) are discrete on the real axis of C with no finite 1imit-point. Proof. Let A = ~ + iv be an eigenvalue of the problem with non-trivial eigenfunction ljJ, then from lemma 4.2 we have A f 0 and from the Dirichlet formula (2.6) b 2 2 [1 J-[OJ b b f {pltjJ' - r~JI + ql'jJl} ljJ ljJ 1 + f M[ljJJ;j;" a
a
a
cot <5'jljJ(b) j2 - cot Y' jljJ(a) j2
on using,
from (2.2) and (2.18),
b - [iApjljJl 2J b + A J S[ljJJ~ a a
tjJ[lJ = ljJ[lJ _ iApljJ, A
and the boundary conditions (4.2), with adjustment if ply by r to give
y
0 or <5
O.
Now mu1ti(4.7)
From (2.10)
f a
b
{;PS[tjJJ - ljJ5hJ}
137
REGULAR DIFfTRFNTUL I;Xf'RI,SS!ONS AND RJ:'LAIT.D OPER,1TORS
i. e. ,
b
im[
f S[IPJ;J;"]
(4.8)
a
From (4.7) taking imaginary parts, and from (4.8) 2 -v(1jJ,1jJ)y,6 _1:11 [rl1jJ12J~ + IAI2 [pl1jJ12]~
o and from (4.6) it now follows that v
=
o.
It has already been shown in lemma 3(b) that all the eigenvalues of the problem (4.1) and (4.2) are simple. Also from lemma 3(b) it now follows that W(x,cp)(a,·) can have only real zeros and so is not identically zero; also that the zeros of W(x,q,)(a,·) are discrete on the real axis of C with no finite limit point. 0 For notation let the zeros (eigenvalues) of W(x,q,)(a,·) be denoted by [An: n EN} where the index set N is empty, or the set {l ,2,3,· .. ,p} of the first p positive integer~, or the_set {l,2,3,····} = N of all positive integers. We write N ~ 00, with N = 00 if N = Nand N < 00 if N is finite. From lemma 4.4 we have A E R (n EN). n
It will emerge from the results in sections 4.9 and 4.10, that for the general symmetric left-definite boundary value problem (4.1) and (4.2) it is always the case that N = 00.
Examples given in section 4.11 show that W(x,q,)(a,·) can be, as an integral function on C, of order 1; thus it is not possible in general, even supposing asymptotic analysis of solutions of (4.1) for large A could be carried out, to deduce that N = from known results concerning integral functions of non-integral order; compare with Titchmarsh [22, section 1.7J. 00
Also for notation let 1jJn denote the (unique up to linear independence) eigenfunction associated with the eigenvalue An' and this for all n E N. Let 1jJn be normalized to satisfy, see lemma 4.3, (n EN).
4.5.
(4.9)
Orthogonality of the eigenfunctions
Lemma 4.5. Let 1jJ and 1jJ be eigenfunctions of the left-definite problem (4.1) and m n (4.2) with Am f An; then (wm,wn\,l\ = O.
Proof. As in the proof of lemma 4.4 the following identities hold {recall An E R (n E N))
(4.10)
w.N. EVERITT
138
(4.11)
(4.12) From (2.10) b
J
a
[2ipIVm;j;"n]~
{;j;"nS[IVm] - IV m SIIVn]J
so that (4.12) may be rewritten as .
_
b
b
(IV m,IV n )y,6 = -[lAnIVmIVn]a + An!
(4.13)
S[IVm];j;"n'
From (4.11) and (4.13) it follows that (recall \n 1 0 (n EN)) -1
Am
) (IVm'~)n"y,6
-1
An
)
(IVm,IV n y,6
and the result follows since Am 1 An' It follows now from (4.9) and (4.10) that {IV n: anal set in Hy,up i.e., (IVm,lVn)y,o
(m, n
= °mn
o
n E NJ E
forms a normal orthog-
N)
(4.14 )
where 0mn is the Kronecker delta. 4.6.
Whas simple zeros.
A more difficult result to prove is Lemma 4.6. For the left-definite boundary value problem (4.1) and (4.2) the following results hold; let 4'>(. ,A) and x(·,A) be the solutions of the differential equation (4.1) satisfying the initial conditions (3.3); let W(\) = W(x,~)(a,A) (AEC) and let W' denote the derivative of Wwith respect to \; then for all \EC (IV ( . ,A) ,~(. ,I)) y,u,= W(A) - AW' (A)
o<
-\W' (\)
y
=
0, 0
1
= ZIT or
1
W(x,~)(a,·)
y
a E
R.
a
"2
IT
/)
=
0
= /) = O.
(4.15)
6 E
(0,
t
IT];
the other two results follow in
From the Green's formula (2.5) for the differential
b
J {;p(. ,I-a)
1
~
are simple (and real).
Proof. We prove the result for y, similar form. Let A E C and expression M
°
= "2 IT,
y
-W(\) - feW' (\)
All the zeros of
y,
_
M[x (. ,Ha)] -
X (.
,Ha) MI~ (. ,X--crl]J
= ext. ,HO)4'>[1] (. ,I-o) - x[1] (. ,Hcr)~(' :r_o)]b
a
= [x(·,i\+a)¢[1]
"A-a
(·.I-a) -
+ [i (A-a)p (.
xfl] A+a
(.,i\+cr);p(.,I_a)]b a
b
>x (. ,A+a );(. ,I-a)+i {A+a)p {'lx(' ,1.+0 );(. ,I-a )]a
REGULAR DIFFERENTIAL EXl'RhSSlONS AND RELATED OPERATORS
139
(4.15)
The left-hand side of (4.16) is also equal to, on using the differential equation (4.1) , b
J "¢(·,i-o) Slx\",Ho)]
(Uo)
a b
f x(- ,Ha)
- (A-o)
$[<1>(. ,I-o)J
a b
[(Ho) - (A-oJJ
f -;p(. ,I-a) S[x(- ,Ha)J a -
-
+ (A-a) [2ip(-)x(·,Ha)<j>(.,>.-0)]
on using the Green's formula (2.10) for S. [1 J [1 J [ x(·,Ha ) ¢J:-a (',\-a) - x Ha b = 20J
b
(4.17)
a
Hence from (4.16) and (4.17)
- b (·,A+a)<j>(·,A-o)]a
-;j;(',i-a) S[X(',Ho)J - 20[ip(·lx(·,A+a)-;p(.,>:-a)]~_
a
(4.18)
Now
x(a'A+a)<j>&:~ (a,i-a) - x~~; (a,\+a)"¢(a,i-a)
W(x,~)(a,A+a)
=
(4.19 )
since ~(a,\) and ~~lJ(a'A) are real and independent of A. Also from (2.26) b
"¢(b,i-a) = ¢(b,\-o) exp[-J J(A-o) where, for all \
a E
C, b
b
exp[-J J(A) = exp[-J (r a
a
r-
-1
(4. 19a)
2iAPP }J
and from (2.28)
¢Il] (b,I-a) A-a
=
¢~lJ A-a
(b,A-o) exp[-JbJ(A-O). a
Hence, recalling that x(b,\) and xf1](b,A) are real and independent of \, [1]
-
x(b,Ha)¢X_a(b,A-O)
-
[1]
--
xHa(b,Ho)¢(b,A-a)
= (X(b,A-O)¢~~~(b,\-o)
-x~~~(b'A-a)¢(b,A-a)}
x
exp[-fbJ(\-o) a
b
= W(x,¢)(b,A-O) exp[-[ ](\-a) a
= W(x,¢)(a,\-a)
(4.20)
on using (2.21). From (4.18), (4.19) and (4.20) we obtain, with
a
f 0,
140
W.N. EVERiTT
-(20)-1 {W(x,q.)(a,>.+o) - W(x,q.)(a,>.-o)} =
f
b
~(·,I-o)S[X(·,Ho)]
_
_
b
- [ip(·lx(·,ic+o)q.(·,A-O)]a
a
Let 0
+
°to obtain (recall W(x,q.)(a,·) is holomorphic on C) b
-W'(>.) = -[ip(.)x(·,A),¢(.,I)]b + f ,¢(.,I)S[X(·,A)] a a
tlT J),
As in the proof of lemma 4.4 (recall Y, 6 E (0, (x(·,A),q,(·,I))
y,
6
= [x[l](.,A)"¢(.,I)]~
+ Af a
b
(4.21 )
~(.,I)S[X(·,A)]
+ cot o'x(b,ic)"¢(b,I) + cot Y'x(a,A)"¢(a,I)
[xfl](">')"¢(''\)J~ - AW'(A) + cot o'x(b,A)"¢(b,I) + cot Y'x(a,A),¢(a,I) on using (4.21).
From the initial conditions (3.3) this gives
[x(·,A),q.(·,I))y,6
=
-xfl](a,;\.)'¢(a,A) + cot Y·x(a,>.)¢"(a,>.) - AW'(A) - x~lJ(a'A)~(a,>.) + x(a'A)¢~lJ(a'A) -AW' (A) W(>.) - AW' (A)
(A E C).
If An is an eigenvalue of the problem (4.1) and (4.2) then An E Rand W(An) = 0; also if ~n is the eigenfunction then from section 3(b) we have (4.22)
x ( . , An) = kn q. ( . , An) with kn f 0; for all three cases of (4.15) this gives kn (¢(· ,An),cp(' ,An))y,O
(4.23)
-An W' (An)'
=
From this result and lemma 4.3 it now follows that W' (An) f 0 so that all the 0 zeros of W(x,cp) (a ,.) are simple (and real). From (4.23) it also follows that _k- l A n
n
W'(A ) n
>
0
and the normalized eigenfunction can be defined by, for all x n E N, ~ (x) = {-k / (A W' (A ))} 1/2 ¢(X,A ) n
n
n
n
(4.24)
(n E N)
n
E
[a,b] and all (4.25 )
where the positive square root is taken. This last result should be compared with the results in Titchmarsh [22, section 1.9J.
141
RliGULAR DIFITRLNTIAL }iXPI
4.7.
The resolvent function q,.
For the left-definite boundary value problem (4.1) and (4.2) we now follow the lead of Titchmarsh [22, section 1.6J and define 4: [a,bJ by using the solutions
~
x
C
x
Hy,o, 7 C
and x and the Wronskian W.
The corresponding formula is given by 4(x,A;f) =
x~(Z)) JX
¢(t,i)S[fJ(t)dt
(4.26)
a
where x
E
[a,bJ, A E C and f
~(A)
E
H
y,~
, and b
=
exp[-f
{r -
r-
2iAPp-l}J
(A E
C).
(4.27)
a
Note that S[fJ E L(a,b) (f E Hy,u,) so that the integrals in (4.26) both exist. Since, in general, the sol utions ¢ and x are not conjugate symmetrical about the real axis (see (2.26)) it is essential to use such terms as ¢(t,i) and x(t,i) which are, however, holomorphic for all A E C. The factor ~(.) in the second term of (4.26) appears in the place shown as a result of taking W(A) = W(x,~)(a,A); if W(x,~)(b,A) is used instead then a similar factor would have appeared in the first term on the right-hand side of (4.26). The immediate properties of 4 are given in Lemma 4.7. (i)
Let q, be defined by (4.26) and (4.27); then q,(x,· ;f) is meromorphic on C, for all x E [a,bJ_ and all f E Hy,o ,
with simple poles at the eigenvalues {An: n and (4.2); (ii)
(iii)
E N}
of the problem (4.1)
q,(. ,A;f) satisfies the non-homogeneous differential equation
M[q,(',A;f)J AS[(p(',A;f)J + S[f] on [a,bJ (4.28) for all A E C \ {An: n E N} and all f E Hy ,6; ¢(. ,A;f) satisfies the boundary conditions (4.2) at the end-points a and b, for a 11 A E C \ {A n : n E NI and a 11 f E Hy,o
Proof. (i) This result follows on inspection of the individual terms of ¢ and the properties of the Wronskian W(·) given in section 4.6. (ii)
This result follows on forming the appropriate quasi-derivatives
¢~OJ("A;f) and ¢~l](.,A;f) and use of (2.21), (2.26) and (2.28); the need for the factor
~(.)
(iii)
appears in these calculations. This follows as in the proof of (ii).
D
142
W.N. EVERITT
4.8.
Properties of
~.
There are certain properties of the resolvent function ~ which are required in section 4.9 for the self-adjoint case, and in section 4.10 for the symmetric non-self-adjoint case. These are given below in a number of lemmas. Lemma 4.8(a). (i)
(i i)
<1>:
-
For all A EO C \ On: nEON} the following properties hold: H -;. H y,o y,o
<1>:
-
H y,o
H y,o
-+
on Hy,o (·,A;f)
=
0 if and only if f
=
(4.29)
O.
For all A EO C \ R there exists a positive number K(A) such that (iii) hr· ,A;f)1I y,o -< for all f EO Hy,u~.
rill -lvl
llfli
(4.30)
y,o
Note that (4.30) shows that may be regarded as a bounded operator on Hy,o to Hy,u"for all AEOC\ R. Proof. (i) This follows from the definition of Hy,u"even though f EO Hy,u,.
<1>;
in general (·,A;f) may be in -
(ii) If f EO Hy,u,and f = a then S[f] = a and ~(. ,A; f) = 0; if f EO Hy, 6 and (·,A;f) = 0 then from (4.28) it follows that S[f] = a and so f = o. (iii) This proof depends on a result of Everitt [11] which we adapt for use here by introducing the additional coefficient r in M through the unitary (isox metric) transformation (Uf)(x) = f(x) exp[-~ r] (x EO [a,b]), as in Everitt [12, sections 4.1 and 5.1]; this yields the following ~ priori estimate. Lemma. Let the coefficients p, q and r satisfy the left-definite conditions; let [a,b] -+ C and a EO L(a,b); then for all E > 0 there exists A(E) > a such that
a:
b
J
a
2
b
lal If I ~ E J plf' - rfl a
2 +
A(E)
b
J
a
qlfl
2
(4.31 )
for all f EO Hy, o. Proof.
See [11] and [12] as quoted above.
Proof of lemma 4.8(a)(;;i). In the proof we use K(A) to represent a positive number depending only on A, but not necessarily the same number on each occasion. Consider (·,\;f) with f EO Hy,o ; following the method used in the proof of lemma 4.4 we obtain (recall S[f] EO L(a,b)) -vII (·,A;f)1I 2 0 y,
b
=
im['.l:'
J a
S[f]]
REGULAR DIFFERENTIAL EXPRESSIONS AND RELATED OPERATORS
i.e. ,
Ivlh(.,A;f)1I
2
6:: y,
143
b
IAI suprl
From the definition (4.26) of
J5i!:lb
1
J
Ivl
a
~
and x on
(x E [a,b])
IS[f]1
where Ivl appears in the denominator to take into account the simple poles of W(·) on the real axis. Hence, for all f E H s' y,u 2 <5 :: K(~) {j b IS[f] 1}2 Y, v a
111;(. ,A;f)1I
(4.32)
Now since p-1 , r, p' and w E L(a,b) and using the Cauchy-Schwarz inequality b
J IS[ f] I
a
b
<
b
K{j If' I + J rip' I + Iwi} If I } a a b
b
:: Kr J If' - rf I + fa If I } a
a b
where
a
:: K[{j plf' - rfl2}l/2 + {j a a = Ip'l + Iwl + Irl E L(a,b}. This gives, on b b J IS[f]I}2 :: KrJ plf' - rfl Z + a a ::
KlIfll~,<5
(f
E
H
b
alfIZ}l/2]
using (4.31) above, b J qlfl2} a
sl.
y,u
The required result now follows from (4.32) and (4.33). Lemma 4.8(b).
(4.33)
o
The residue of
kn b AnW'(A n ) ~(X,A n )[(f,~(.,A)) a n y, <5 + riA npf~(·,A n PI]
for all x E [a,b} and all n E N; here kn U 0) is given in (4.22), i.e., x(· ,An) = kn ~(. ,An) on [a,b]'
(4.34)
(4.22)
Remark. The formula (4.34) well illustrates one of the difficulties in the general case of the boundary value problem (4.1) and (4.2). The presence of the integrated term iA npf~(·,A n ) prevents the direct use of the classical methods in Titchmarsh. Proof.
From (4.22) we obtain x(a,A n ) = kn~(a'An) and from (2.26) we have (recall
A E R) n
b
i(a,A ) = x(a,A ) exp[J ](A ) n nan in the notation of (4.l9a); thus if ~(a,A ) f 0 b n
kn
=
kn exp[J ](A ) a
n
(n
E
N).
(4.35)
144
W.N. EVERITT
If ¢(a,A n ) = 0 then ¢fl](a,A n ) f 0 and a similar argument involving derivatives shows that (4.35) hold~ in all cases. From the definition of ¢ in (4.26) and since, from (4.24), the pole at An is simple with W' (An) f 0, the residue of ¢ at An is x b kn ¢(x,A n ) ¢(t,An)S[f](t)dt + ~(An)kn ¢(X,A n ) [ ¢(t,An)S[f](t)dt
!
divided by An' From the definition of ~(~) in (4.27), the notation (4.19a), and (4.35) it follows that ~(A n )kn = kn (n E N) so that the residue is b
k {W' (A )}-l ¢(X,A ) J "'¢(t,A )S[f](t)dt. n n nan
(4.36)
Now from (2.10) (reca 11 An f 0) and (4.1) b
{2 i p f¢' ( . , An ) } I~ +
J ¢'(t,An)S[f](t)dt a
i f S[ ¢( . , An )] b
b
A- l [{2iA pf¢,(·,A )JIb + J f M[¢(·,A )] n n n a a n
(4.37)
Also, as in the proof of lemma 4.4, b
(f ,¢ ( .
, A ))
n
y,
6 =
{i \ pf'¢ ( . ,A n
n
)}
Ib + J f M[ ¢ ( . a
a
,A )] n
(4.38)
The required result follows from (4.36), (4.37) and (4.38). We note that if pta) = 0 or of wat An takes the form
y
D
= 0, and p(b) = 0 or 6 = 0, then the residue (xE[a,b]
nEN)
(4.39)
where ¢n is the normalized eigenfunction. Lemma 4.8(c). i.e. ,
The resolvent function ¢ satisfies the Hilbert relation on H
6'
'I'
¢(X,A;f) - W(X,A';f) for all x E [a,b], all A,
(A - A' )¢(X,A;
(4.40)
n n E N} and for all f E Hy,o Proof. We note that the solution (ii) and (iii) of the non-homogeneous boundary value problem in lemma 4.7 is unique; for if there were two such solutions their difference would yield a solution to the homogeneous boundary value problem (4.1) and (4.2); such a solution must be the null function since A ~ {An: n EN}. A'
E C \ {\
Consider the left-hand side of (4.40); we have M[ ¢ ( . ,A; f) - ¢ ( . , A' ; f)] = AS [w ( • , A; f) - ¢ ( . ,A' ; f)] + S[( A-A' )
Thus Y = ¢(. ,A;f) -
<1>(.
with F
M[V] = AS[V] + S[F] ,A' ;f), where V also satisfies the boundary conditions; in
(A - A'
)<1>('
,A' ;f) is the solution of the non-homogeneous problem
REGULAR DIFFT:RHvTIA L L'XJlRYSSIONS AND Rlc'L/\ TED OJl1:'RATORS
view of the uniqueness result this implies that Y = resul t.
l
~(.
145
,A;F) and this gives the
o
Now let an o~erator R)..: H > H be defined by, for all 1',6 1',0 E C \ fA: n E N} n (R/)(x) = ~(x,A;f) (xE[a,bJ f E H). y,iJ
(4.41 )
From the resu lts in this section we have (i ) (i i)
RA is a bounded operator on H1',0 RA f = 0 if and
f =0
only if
(4.42)
R - R , = (A - ),' ) R), R , . A l A If (R A: ), E C \ RJ is to be the resolvent family of a self-adjoint operator in H1',6 two additional properties require to be satisfied ( iii)
(i )'
R,: H A
,
1',6
-+
H
~
1',,,
(4.42a) where R~ is the adjoint ope~ator of R\. Such a family {RAJ determines uniquely a self-adjoint operator T in Hy,u~ by means of the formula
T=AE+R~l -1
for any choice of A E C \ R; here E is the identity operator on Hy,o' RA exists from (ii) of (4.42), and it may be shown that T is independent of the choice of , E C \ R in (4.43). These results are taken from Akhiezer and Glazman [1, section 75J. It is not possible to prove (i)' and (iv) of (4.42a) for (RAJ of (4.41) in the general case of the symmetric boundary value problem (4.1) and (4.2); examples will be considered in section 4.11. Since it is important for the next section we give here a result on the form of proof of (iv) of (4.42a), which does hold in the self-adjoint case to be considered in the next section. Lemma 4.8(d). Le! the operator R, be d~fined by (4.41); then for all ), E C \ {An: n E N} and for all f, 9 E H y,8 (R/,g)y,6
{-iApR\f·9}1~ + _
b
A
(f'Rxg)y,o = {-iApf.R~}la + A
f
b
S[R)/Jg +
f
b
S[f]g
(4.44)
a
a
!
b
f5(R~J +
f
b
f5[g1 a Proof. Follow again the method of proof of lemma 4.4, recalling that R), f = ~(. ,A;f) satisfies the boundary conditions (4.2).
(4.45)
0
In general it is impossible to prove that the right-hand sides of (4.44) and
146
W,N, EVERJTT
(4.45) are equal, and it is this which prevents (iv) of (4.42a) holding in the general symmetric case. However it will be shown in section 4.10 that it is possible to draw some positive conclusions in the general symmetric case from the form of the results i n 1emma 4. 8 ( d ) . We conclude this section with two more properties of the resolvent function
Lemma 4.8(e).
If
as defined in (4.26), is written in the form ¢(x,\;f)
f
=
b
K(x,t;\)S[f](t)dt
(x
E
[a,b])
a
where K(x,t;\)
x(x,\)¢(t,~)/W(A)
(a ~ t ~ x ~ b)
~(A)
(a ~ x :: t :: b)
= =
then K(x,t;A) = K(t,x;\) Proof.
(x, t
E
[a,b]).
This follows on a calculation using, in particular, (2.26) and (2.28).0
Note that thi s symmetri ca 1 form of K does not, in general, carryover to
<jJ
or RA; in particular it does not entail property (iv) of (4.42a) for R\. Lemma 4.8(f). The eigenvalues (An: n E N) and eigenfunctio~s (1Pn: n E N) of (4.1) and (4.2) satisfy the integral equation, for all n E N, ~n(x)
= An
f
b
K(x,t;O)S[~n](t)dt
(x E [a,b]).
a
There are no other solutions of this integral equation. Proof.
the and the the 4.9.
We have, from (4.1) and (4.28),
M[ ~ n - \ n¢ (. ,''t'n 0 . ", )] = \ n5[~ n] - AnS[1jJ n] = 0 0 n [a , b] ; term in [ ... ] on the left-hand side satisfies the boundary conditions (4.2) so must be null on [a,b] since 0 is not an eigenvalue. If A and 1jJ satisfy integral equation then M[1jJ] = AS[1jJ] on [a,b], from (4.28), and 1jJ satisfies boundary conditions; if ~ is not null then A E {A : n EN}. 0 n
The self-adjoint case.
one additional condition on the left-definite boundary value problem the difficulties discussed toward the end of the previous section can be overcome, in order to represent the problem by a self-adjoint operator in Hy, 6 \~ith
We state Theorem 4.9. Let the left-definite boundary value problem (4.1) and (4.2) satisfy the basic conditions (2.1) and (2.7); let condition (3.20) or (3.21) hold;
147
REGULAR DIFFERENTIAL i:XJ'RJ:SSIONS AND RELATED OPERATORS
additionally let the following self-adjoint condition be satisfied (i)
pea)
0 or
I'
(ii)
pCb)
0
or
6
0
=
(4.46)
0;
let the resolvent function ¢ be defined by (4.26) and (4.27); let the resolvent operator be defined by (4.41); then (1) the resolvent family {R A} satisfies the following results for all A,A'EC\R - and is a bounded, compact operator on H (i) R: H ~ H A 1',0 y,o y,8 (i i) R,f = 0 if and only if f = 0 in H 1',8
A
(i v) (v)
(2) (i) (i i)
R~
R"A
IlRAIl .::.. \v\-l where A = the operators {R : IJ
IJ
IJ
(4.47)
+ iv
E R \ {,\ : nEoN} satisfy the following results n
RII : Hy,u£ ~ Hy,v£ and is a bounded, compact symmetric operator on H1',6 R- l exists and is an unbounded, self-adjoint operator in H ~. y,
II
v
If the operator T is defined by -
OCT)
{RAf: f E Hy,fj}
Tf=Af+R:lf
(f
E
OCT))
(4.48)
-
for all A E C \ {An: n EN}, then T is a~ unbounded self-adjoint operator in Hy,o' T is independent of A, OtT) is dense in H and, in particular, 1',6
T=R~l;
(4.48a)
T has a discrete, simple spectrum given by otT) with corresponding eigenvectors
= {An:
n
(4.49)
N}
E
n EN}, and N = N =
{~n:
roo
The self-adjoint operator T in Hy,us represents the left-definite boundary value problem (4.1) and (4.2). Proof. See below. Corollary 4. 9(a). Let f E H and write (uniquely) f 1',6 hE H 8 H ,then y,o y,o (h ,
\
'Y,'
0
(n
E
g + h where g E Hand y, 0
(4.50)
N)
and if cn
(f,~) n y, ,\
(g'~n) y, v£
(n
E
N)
(4.50a)
148
[!'N EVERITT
then 1/
Proof.
I
gil 2
y,6
2
(4.50b)
n=l Icnl .
See below.
Corollary 4.9(b).
Let f E D(T) then f(x)
=
I
c
nO:l
n n
(x
(x)
E
[a,b])
(4.50c)
where the series converges absolutely and uniformly on [a,bJ. Proof.
See below.
(1)
We first prove the results stated in (4.47).
(i) Let f E H,(, 6' i.e., (f,g) y, 6 = O(g E Gy, 6) with G,(, 6 defined in (3.24); then, with w defined in (4.26) and following the method used in the proof of lemma 4.4, we have, on using the self-adjoint condition (4.46), b
(w(. ,A;f),g)y,O = {iApw(' ,A;f)gl I~ + f M[w(' ,A;f)]"9 a
b =
f
M[1i(' ,A;f)]"9
a
(from (4.46»
b
=
A J S[
Af a
b
w(· ,A;f)S[g]
+
b
J a
f
b
(g
fS[g]
a
o from (2.10, (4.46) and 5[g] = 0
S[f]g
(g E
G
J;
thus 1i(.,A;f)
y,o
G ,)
E
y,u
E
H " y, 0
It w~s shown in lemma 4.8(a)(iii) that RA is a bounded operator on Hy ,6' and hence on Hy,u" The proof that R, is a compact operator on Hy,6 follows from the definition of
(in
This result follows as in lemma 4.8(a)(ii).
(iii) This result (the Hilbert relation) was proved in lemma 4.8(c). ( iv) The proof of this result was essentially started in section 4.8; from (4.44) and (4.45), with the addition now of (4.46), we have for all f, 9 E Hy,6 b (4.52) (RAf,g) 6 - (f,Rx-g ) 6 = A f {S[RAf]g - fS[L9J} y,
y,
a
A
where we have called on (2.10) and (4.46) to remove the other terms. results also give
These
149
REGULAR DIH'nRl'NTL1L I'XPRf'SSIO.\!S AND Rf'LATf'D OPI:RATORS
b
J {R\f.S[g] - S[f]·Rx9} a b
J
a
b
R\f{M[R~]
\S[R-9]} \
\
J {M[R\f]
a
b
J {:;g)]- M[
- AS[RAf]}R~ ~(.,>:;g)}
a
=0
(4.53 )
from (2.5) since, under (4.46),
rJ>~l](.,A;f)
=
and
(f,Rx9)y,o
=
(f, g
E
H,) y,u
Hence (4.54 )
and this gives the required result. We note that (4.54) gives an alternative proof of the boundedness of the operator _ RA (see lemma 4.8(a)(iii)); RA and RA are both defined on the whole space H ,and the boundedness of R, and R_ follows from the general result given y,u A A in [1, section 28, page 82]. (v) This inequality for the bound of R\ follows from the remark made in [1, section 75, footnote to page 251]; in fact from the Hilbert relation (iii) above, with A' = >:, 12vIIIRAR~1I
12v I" R 112 A
IR - RI A A :: "RA" + IIR~II <
(4.54a)
211RAII
and this gives the required inequality. For use in the next section 4.10, in order to apply the Titchmarsh analysis of [22, chapter 2], it is essential to give a direct proof of the inequality, for all A E C \ R, l HII h(· A·f)1I < Ivl(4.55 ) (f E H J " y,o y,o y, u which is equivalent to the inequality (v) of (4.47). As in the proof of lemma 4.4, with repeated application of (2.10) and calling on (4.46), b
(
y,
r
u
-
(f,rJ>(· ,\;f)) y, ,u
J {M[
- fM[]}
a
f
b
{\S[CP]f - ifS[CP]}
a
\ f
b
b
rJ>S[f] - >: J S[f]~ a a
(from (4.28))
150
W.N. EVERITT
b
=
i-
b
J
J {M[¢]
- "\Sh]l - i
a b
=
b
AJ ¢M[ ¢] - i J M[¢]~ + a a b
= A
- AS[¢]}¢
a
J ¢M[ ¢]
- i
a
b
J
I A 12 {2i pi ¢ 12} I~
~M [ ¢ ].
(4.56) (4.57)
a
Also, using the same methods and again calling on (4.46), ( ¢ ( . , A; f) , ¢ ( . , A; f) )
b _
Y,
_
=J
6
a
2
b
} Ia
(4.58)
b
J ¢M[ 1l]
(4.58a)
a (¢ ( . , A; f) ,1l (. , A; f))
y,
6 =
=
f
b
a
f
¢M[ ¢] -
(4.59)
¢M[¢]'
(4. 59a)
b
a
From (4.57) and (4.58a), (4.59a) we obtain (A - i)(¢(·,A;f),¢(·,A;f))
~
y,u
(¢(.,A;f),f)
s
Y,u
-
(f,
~
Y,u
i.e. , 12vlh(·,A;f)11
2 < 2 I1 i1>(·,A;f)11 HII y,6 . y,8 y,o
and so II
¢ (. ,
A; f) II
~
<
y,u -
Iv 1- 1
HII
y,6
(A E
C \ R; f E
Hy,vs)
(4.60)
and this is (4.55) again. This completes the proof of (1) of theorem 4.9. ~2)(i) No~ let )1 E R \ {An: n EN}; then R\l is well-defined by (4.41) and maps H s into H ~ as in (1 )(i) above; the compactness of R also follows from y,u y, \l the form of the integral operator. The symmetry of R follows the same proof as )l in (1 )(iv); the results (4.52), (4.53) and (4.54) continue to hold when A = \l E R. The boundedness of R then follows from the general result in [1, section 28]; \l however there is no estimate for the norm of R since (l)(v) is no longer valid \l when A = )l is real. (ii) The inverse R- l exists as in (l)(ii) and is a self-adjoint operator in \l Hy ,6 from the resul! in [1, section 46, theorem 3J; the dom~in D(R-l) = {R f: f E H } and this is strictly contained in H and so, since R- l )l )l y,6 -1 y,6 )l is closed, a contradiction would result if also R)l is bounded. U
The definition of the self-adjoint operator T by (4.48) is taken from the result given in [1, section 75, footnote to page 251]. In view of the properties of R given above, and from the Hilbert relation (4.40), it may be shown that, \l where E is the identity operator,
151
REGULAR DIFFERENTIAL jCXPRESSIONS AND RELATED OPER.rrORS
(T - IlE)R 11 f = f (f E H) R11 (T - IlE) f = f (f E O(T)); (4.61) y,o 1 exi s thus (T - IlEf _ ts and is a bounded, symmet ri c op elator on Hy,o> for all jl E R \ {\ : n E NL Hence the definition (4.48) of T extends to all n 1 A E C \ {\ : n E NJ and, in particular T = Rto give (4.48a). This shows that T n
0 _
is an unboun~ed self-adjoint operator so that OtT) is_dense in HY,Q but is not the Also the spectrum otT) -C {A n : n E N} so that T has a discrete whole space Hy,us spectrum. Suppose now" is an eigenvalue of T with eigenvector ~, i.e., T~ = A~; then " is real and ~ = RoT~ = "Ro~ and so the differential expression M can be applied to \jJ to give M[Iji] = AM[Rolji] = "S[~] on [a ,b], i.e., (4.1) is satisfied; also ~ satisfies the boundary condition (4.2) at a and b. Hence A oo" n for some n E N and
~
= ljin'
On the other hand if "n is an eigenvalue and ~n an eigenfunction of (4.1) and (4.2) then M[~ n] = " nn \jJ ; also M[" R ~ ] = A S[~ ]; hence M[~ R ~ ] = 0 on non n n n -A non [a,b] and ~ - A R \jJ satisfies the boundary conditions at a and b; since 0 is not n non an eigenvalue ljin - AnRo~n = 0 on [a,b] and ~n E OtT); this gives Tljin = "nTRo\jJn = An\jJn; hence An is an eigenvalue and ljin an eigenvector of T. Since now otT)
=
{\
n
-
-
n E N} and T is unbounded it follows that N = N =
:
00,
Taken together it is clear that these results imply that the self-adjoint operator T represents the boundary value problem (4.1) and (4.2) in the chosen Hilbert function space Hy,us'
o
This completes the proof of theorem 4.9. Consider now corollary 4.9(a).
Let f = 9 + h as shown, i.e., hE Gy ,/) and S[hJ = 0 on [a,bJ; then, as in the proof of lemma 4.4, for all n E N, (h,lji) n
,,= af y,
b
b
"nfa n ~ S[ h]
Mh]h n
O.
The Parseval identity (4.50b) holds as a special case of the spectral representation theorem for self-adjoint operators with discrete spectrum. However it is important to note, for use in the next section, that (4.50b) may also be proved using the classical methods of Titchmarsh [22, section 2.l2J. Define the analytic function 1Ji by
= (1)(·,,,;f),f) y,us
n E N}' f E H ) (4.62) n y,o then 'Ji is a meromorphic function on C with simple poles at{" : n EN}; the residue n of 1Ji at An is given by, see (4.39), -lcnl2 (n E N) where c n = (f,IV) 6(n EN). n y, From the Hilbert relation (iii) of (4.47), with \' = 0 and f = Tg for g E OtT), <Jl(x,A;Tg) - 1>(x,O;Tg) = H(X,\;1>(' ,0;Tg)); 1Ji(,,)
(\ E C \
{A :
W.N. EVERITT
152
i.e., for 9
E
O(T), for all x
E
[a,b], and for all ).,
(X,A;g) = A-
l
C\ R
E
{-g(x) +
(4.63)
The proof of (4.50b) now follows identically the argument in Titchmarsh [22, section 2.12) where (4.63) replaces [22, lemma 2.9) and (4.47) replaces [22, lemma 2.8]; this proves (4.50b) on the domain OtT) and the extension to D(T) = Hy,Q o follows the argument in [22, section 1.13]. We note that the Parseval identity (4.50b) implies that 00
g =
I
n=l
c >jJ n n
with convergence in the norm of Hy,us again follows that N = N =
(g
E
H
(4.64)
s)
y,u
Also, since Hy,u"has infinite dimension it
00.
Consider now corollary 4.9(b). The proof of this result follows from an identical application of the result in Titchmarsh [22, section 2.13J. Remarks (i) The case when p(x) 0 (x E [a,bJ) is included in the self-adjoint case (4.46); in particular this covers the left-definite case of the symmetric differential equation
M[Y] = ).,wy on [a,b]
(4.65)
with symmetric boundary conditions (4.2), when w is not null on [a,b] but can be of arbitrary sign and can vanish on a set of positive measure within [a,b]. (ii) We have not invoked the theory of compact operators in the proof of the results of this section but this does provide an alternative method to prove theorem 4.9; see the results given in [3, chapter 7J, or, in particular, the account in Taylor [21, section 6.41) which is appropriate to the results considered in this paper. 4.10.
The symmetric case Consider now a return to the general symmetric case of the boundary value problem (4.1) and (4.2) but now without the self-adjoint condition (4.46). It is shown in this section that much less can be proved in this case, in comparison with the self-adjoint case of the previous section; however it can be shown that the problem always has a strictly countable number of eigenvalues, i.e., N = 00, or, equivalently, that the Wronskian W(x,~)(a,·) has an infinity of zeros (all real and simple). It does not seem to be known in this case, however, if the eigenfunctions {>jJn: n E N} span the whole space Hy, us' or even if the projection _ of the {>jJ n : n E N} into the reduced space Hy,us has a linear hull which is dense _ in Hy,us' -
We start by introducing two sub-spaces of the space Hy ,6:
153
RECCL.4R DlFFhRhNTL4L J;X.l'RLSSIONS AND REL/I'lFD OPERAFORS
(i)
Let the sub-space o Hy,o of
Hy,o be defined by Hy,us: f(a) = f(b) = OJ
{f E (4.70) oHy,o noting that the restriction is unnecessary if y = 0 or 0 = 0; if y = 0 = 0 then 'but , otherwise OHy 0 is a strict sub-space of Hy,us; however it is not oH0,0 = H0 0 · difficult to see that Hy,o e 0 H y,o can be at most two dimensional (indeed a basis for this space can be constructed from the two solutions ¢(. ,0) and X(· ,0) of (4.1)); thus we have = co; (4.71) dim oHs y,u
(i i) Let ¢ : [a,bJ + D(M) be defined as the unique solution of the nono homogeneous boundary value problem
M[yJ = S[¢(· ,O)J on [a,bJ
(4.72)
where cp(. ,0) is the solution of (4.1), with tions (3.3). Similarly let Xo: [a,bJ
+
= 0, defined by the initial condi-
A
D(M) be defined as the unique solution of
M[yJ = S[x(· ,O)J on [a,bJ
(4.73)
Standard existence theorems, suitably extended to the case of the genera li zed differential equat ion (4.1), show that ¢o' Xo exist, and since ¢o' Xo E D(M) it follows that
oHy , Ii
e
{q, 0
'
Xo} = {f
E 0
Hy , 0:
(f, ¢ 0 \
6
'
=
(f, Xo \, 6
= D},
(4. 74)
i.e., this subspace is oH s reduced by the two-dimensional subspace generated by y, u • the projection of
(4.78)
J f(t) M[xoJ(t)dt
again using f(a) b.
a 0; the required result now follows at a.
fib)
Similarly at
-
Secondly we prove that
E
Hy,u,; for let g E Gy,6 , then from (4.77)
b
(
J M[
,
y,u
(4.79)
a
b
J S[f] a
9
b
= f f 5[gJ
(since f(a) = fib) = 0)
a =0
(since g
E
G
,).
y,o
Lemma 4.l0(b). Let the function
Ivl-llltil
<
y,6 -
y,o
(f
E
H
o y,6
).
(4.80)
Proof. This result is of the same form as (1 ltv) of theorem 4.9; note however that, in the symmetric case, (4.80) holds only, in general, on oH 0 as opposed, 'I, in the self-adjoint, to (4.47)(v) holding on all of H o. y, It is not possible to prove (4.80) by the operator-theoretic proof as used in establishing (4.54a) in the self-adjoint case, since
-
oHy,o <jJ (
e
x, 0; f )
{¢o,xo } (x
E
[a, b], fED (Ro ) ) ;
(4.81 )
then _ Ro: D(R 0 ) + oH y, u"from (4.76); also Rof = 0 in o Hy, <5 if and only if f = 0 in H " i.e., f(x) = 0 (x E [a,b]) since, as before, M[R of] = S[f] on [a,b]; o y,u 1 hence the inverse operator R- exists. o
155
RIlCULAR DIFFERENTIAL EXPRESSIONS AND RFL4TllD OPERATORS
However, it is important to note, and the difference here between the selfadjoint case and the general sym~etric case is significant at this stage, that Ro is not a symmetric operator in 0 Hy,u,8 {¢ 0 ,x 0 } since it maps this space into the ~ larger space oH y,(\ It is now possible to define the operator Q by O(Q) and
Qf
then
Q: O(Q)
C H \ - 0 '1,'
onto
f
E
O(Ron
(f
E
O(Q));
-
H
k
0 y,u
e
(4.82)
{q, ,x }. 0
0
It wi 11 appear in the next section devoted to examples that D(Q) mayor may not be dense in oHy,o however since the mapping by Q is onto it follows from (4.75) that dim(D(Q)) (4.83) 00.
Lemma 4.10(c). Let f E O(Q); then (i) for all A E C \ R - l !If II II ( . , A; f) II y, c5 -< Ivl y,o l II ( . , A; Qf ) II < II QfII '(,6 - Ivl y, I) l (x,A;f) = A- {-f(x) + ¢(x,A;Qf)} (x E [a,b]) (ii)
(4.84) (4.85)
the residue of (x,· ;f) at the simple pole An is given by (x
E
[a,b]).
(4.86)
Proof. (i) The two results in (4.84) both follow from lemma 4.1o(b) since both f and Qf are in H k· o Y.u The result (4.85) follows from the Hilbert relation (4.40) with A' 0, f replaced by Qf and the properties given above of the operator Q. (ii) This follows from the results (4.34) and (4.39); note that the integrated term in (4.34) vanishes since f(a) = f(b) = 0 with f E 0(0). 0 We now pass to the main result in the general symmetric case. Theorem 4.10. Let the left-definite boundary value problem (4.1) and (4.2) satisfy the basic conditions (2.1) and (2.?); let condition (3.20) or (3.21) hold; let {An: n E N} and {o/n: n E N} be the eigenvalues and ortho-normal eigenfunctions of the problem as given by the zeros of W(x,q,)(a,·), and (4.14) and (4.25); for any f E Hy ,6 let {c n : n E N} denote the generalized Fourier coefficients (n -
-
E
N);
(4.87)
let the sub-space H 6 of Hy, 6 be defined by (4.70); let the operator Q with _ 0 '1, domain D(Q) -C 0 Hy,
W.N. HVERITT
156
(f
E
0(0)),
(4.88)
and with convergence in the norm of oH y, 0 f =
Proof.
I
nEN
c
(4.89)
n
See below.
Corollary 4.10(a). Let all the conditions of theorem 4.10 hold; then N, the number of eigenvalues of the problem, is strictly countable, i.e., N = in consequence the integral function W(x,¢)(a,') has an infinite number of zeros (all real and simple). 00;
Proof.
See below.
Corollary 4.10(b). then
Let all the conditions of theorem 4.10 hold; let f
E
0(0):
00
f(x) =
I
n=l
c 1jJ (x) n n
(x
E
[a,bJ)
the series being absolutely and uniformly convergent in C. Proof of theorem 4.10. all f E 0(0), ~(A)
=
As in (4.62) defire the meromorphic function (
~
by, for
-
then ~ is meromorphic on C with simple poles at {An: n E N} only with residues -Ic 12 (n E in, on using (4.86). The proof of (4.88) now follows, identically, n the argument i~ Titchmarsh [22, section 2.12J where (4.85) replaces [22, lemma 2.9J and (4.84) replaces [22, lemma 2.8]; this gives the required result on 0(0); the extension to the closure 0(0) follows from the argument in [22, section 1.13J. The result (4.89) then follows from standard arguments in Hilbert space theory. 0 Proof of corollary 4.10(a). This follows from the Parsev3! identity and the result dim(D(O)) = of (4.83). 00
0
Proof of corollary 4.10(bJ. This follows from a'l appropriate appl ication of the method used in [22, section 2.13J. 0 Remarks. (i) It is to be noted that theorem 4.10 has nothing to say concerning {1jJ n: n E N} into the sub-space Hy,u,B Oy,u H ,. the projection of the eigenfunctions _ end this because 0(0) -c 0 Hy,u,i unlike the self-adjoint case, see corollary 4.9a, it may happen in the symmetric case that (h,1jJ) n y,u, f 0 for h E Hy,u,8 0 H y,u,or even h E Gy,o (ii) Even if 0(0) = oHy,o no information can be drawn from theorem 4.10 concerning the completeness of the eigenfunctions {1jJn: n E N} in the space Hy,o; the only examples available, see section 4.11, either give {1jJn} complete in Hy,o
157
REGULAR DIl-'l+RENTI.'l L L'(PRESSIONS AND RELATED OPERATORS
or are indeterminate due to technical difficulties; it seems to be an open question as to whether or not {~n} is complete in o Hy, 6' in Hy, 8' or in Hy, o' (iii) Another open question would seem to concern the order of W(x,~)(a,·) as an integral (entire) function on C; all the examples point to the order being 1 if p is not null on [a,bJ, and ~ if p is null on [a,bJ; if this is the case then no information concerning the number N of eigenvalues can be obtained from the theory of integral functions when p is not null on [a,bJ. (iv) The results of this chapter are largely due to appropriate applications of the methods of Titchmarsh [22J; it would be of interest to know if an operatortheoretic proof of theorem 4.10 and corollary 4.10(a) can be given. 4.11.
The left-definite case-examples.
We discuss a number of examples to illustrate the results given earlier in this section. Example 1. Let a also y = 8 = 0, i.e.,
0 and b = 2n, and p = 1, q
-y"= iAY' on [0,2nJ
and
y(O)
w = r = 0, Y(2n)
=
=
p
= ~ on [0,2n];
0;
note that case 1 of section 3 holds, in fact (3.20) is satisfied. This is an example of the self-adjoint case considered in section 4.9. {I} and it may be seen that G = {OJ 0,0 H0,0 = 0 H0,0 = {f E AC[O,2rrJ: frO) = f(2n) = 0, f' E L2 (0,2rr)} with 2n
(f,g)O,O
J
f'
Here
gr.
o
A calculation shows that ~(X,A) =
1
1 - e iA(2rr-x) iA
and (Ic
E
C).
We note that W(x,~)(· ,A) is an integral (entire) function of order 1 with zeros at the points An = n (n = ~l ,~2, ... ); note also that 0 is not an eigenvalue (see lemma 4.2). The normalized eigenfunctions of this problem are, for n = ~1,~2, ... , ~n(x) = n- I (2rrr l / 2 (1 - e- inx ) (x E [0, 2rrJ). It
may be seen directly that
1 2n . . .1,) = -- J e- 1mx e 1nx dx = 6 . m'o/n 0,0 2IT 0 m,n Either from theorem 4.9, or directly, it may be seen that the set
(~
{~n}
is complete
WoN. EVERITT
158
in H Note that, in the sense of the inner-product in H0,0 , the function x on 0,0 [0,211] is orthogonal to all the {~n} in H0,0 ; however x f/; H0, 0 since this function does not vanish at the end-point 211. It is possible to calculate the form of the operator T of section 4.9 in this case; we find X 211 (R f)(x) = <Jl(x,O;f) " {(x - 211) J tif' (t)dt + x J (t - 211 )if' (t)dt} o 11 0 X
i-
for all x E [0,211], and from this a calculation shows OtT)
=
{g: [0,211] ...,. C 9 and g' EAC[0,211]' gil n 2 (0,211), g(O) 1
g(211)
OJ
and (x E [0,211]).
(Tg)(x) " i (9' (x) - g' (0))
It has to be regarded as exceptional that we can calculate T explicitly. Note that T is obtained from a differential expression of the first-order; formally this is in line with writing T = (M-1S)-1 even though no definite meaning can be given to the right-hand side. Example 2. 0,6
y
=
~11'
Let a " 0 and b " 211, and p
=
1, q
i.e., -y" " iAY' on [0,211] and Y(O)
=
21 on [0,2 11 ]; also y'(211) + ~ iW(211) = O.
w = 0,
= 0,
p "
This is an example of the symmetric, non-self-adjoint case of section 4.10; here again Gp ,W = {l} but G0, 1"ZiT = {OJ so that H0 , 1'ZTI = H0 ,''2TT 1 with 2 H ,1 = {f E AC[0,211] 1 f(O) = 0, f' E L (0,211)} O 211 and H = {fEH If(211) = OJ; o 0, ~iT 0,'211 also 211 (f,g)o , "11 = J f' go. 2 0
-
1
I
Here cp(x,A)
- e -iAx
and W(X,CP)(O,A) =
t
(1 + e
iA211
)
which again is an integral function of order 1. The eigenvalues are given explicitly by An = n + ~ (n = O,~l ,~2,···) which are strictly countable in number, see corollary 4.10(a); note again, see lemma 4.2, that 0 is not an eigenvalue. The normalized eigenfunctions are given by (x
o
H
A straightforward argument shows that the set but al so in the whole space H 1 = H"
O,~TI
O,YzTI
O,~TI
{~n}
E
[0,211]).
is complete not only in
159
RDCL'LAR DIFFliRliNTIAL LX1'RESSIONS AND Rf:L41'liD OPERATORS
We have also
f
~(x,O;f)
x
f
2TI
(x E [0,2 TI ]). f' (t)dt o x o and this follows directly from Since y = 0 we have already ~(O,O;f) However the above; thus the reduction (f,x) I = o is automatically satisfied. o 0, ~2TI 2 reduction (f,~) I = 0 is necessary; it may be seen that ~ (x) = x /(2i) o o 0 ,'2TI (x E [0,2TI]) and so 2TI 2TI if f(t)dt; f tf' (t)dt (f, ~o)o , l<2TI = o o thus (Rof)(x) =
OHO,'iTI 8 {
tf' (t)dt + ix
AC[0,2 1T ] I f(O) = f(2 1T ) = 0, f' 21T and f f(t)dt = OJ.
E
E
L2(0, 21T )
o
In the notation of section 4.10 {R f I fED (R ) =
D(Q)
o
0
H , e {
0 0, 21T
{g: [0,21T]>- C I 9 and g'
0
E
AC[O,21T], g"
E
L 2(O,21T)
and g(O) = g(21T) = g' (0) = g' (21T) = OJ ig' (x)
(Qf) (x)
(x
E
[0,21T]).
It is not difficult to see that the closure D(Q) is oH0, 1'i1T which, from theorem _
4.10, confirms the statement made above that the set Example 3. Let a = 0 and 1. a 1 so y = 6 = 2TI, 1.e., _yn
=
b
{~
n
} is complete in
= 21T , and p = 1, q = w = 0,
p
H
1
0 0, '2TI
= ~ on [0,2 1T ];
iAy' on [0,21T]
This is an example of the pathological case (2) of section 3, i.e., (3.22) hol ds. Here 2TI
and (f,g )]2 TT ,~TI
f
f'
9'
0
with null element the class
(l l·
For this example
= ~(l
+ e- iAX )
iA X(X,A) = ~(l + e (21T-x))
and 1.1A (e21Ti A - 1 ) . W( x,<jl ) ( O,A ) = 4 Note that, as required by lemma 4.2, 0 is an eigenvalue of the problem; in fact 0 is a double zero of 1-/; there are simple zeros at {~l ,~2, ... l; compare with lemma
W.N. EVERITT
160
4.6 which holds when case (1) of section 3 is satisfied. In spite of the double zero at 0 there is only one linearly independent eigenfunction at the eigenvalue 0, i.e., ~o(x) = (2n)-1/2 (x E [0,2nJ); the other eigenfunctions are given by ~n(x) = n- l (2nf l / 2 (1 + e- inx ) (x E [0,211J). However note that ~o is actually the null element of H,fIT ., ,2'TT even though it is a non-trivial solution to the boundary value problem. The eigenfunctions {~ n : n = O,~l ,~2,.·.} are not complete in. H,/211., 1.2TT ; the function x, which we note is the second solution of the differential equation with A 0 but does not satisfy the boundary conditions, is a member of H,"'zIT ., ,", 2Tf is not the null element but is orthogonal to all the {~n}' Example 4. In this final example we show that, in the symmetric non-selfadjoint case of section 4.10, it is possible for the closure 0(0) of the set 0(0) , not to be the whole space oHY,u". Whilst this does not prove that the closure of the eigenfunctions {~ n} is not the space 0 Hy,u< or Hy,u< or Hy,u<' it does show that the methods used to prove theorem 4.10 will not suffice to prove completeness of the {~n} in oHy,o" Let 0
<
a
p =1 1et y, 8 E
<
b
<
ro
and
q=r=w=oon[a,bJ,
p(x) = x- 4 _ 1
1 (o'zn] but exclude the case y = 0 - Z 11. H {f E AC[a,bJ If' E L2 (a,b)) y,o
(xE[a,b]);
We have
b
(f,g) y,v-' Gp,w
J f'g'
+ cot y·f(a)g(a) + cot o·f(b)g(b) a {f E AC[a,b] I (pf)' + pf' = 0 on [a,b]}
{{l}}
H
o y,o
{fEH
y,o
If(a)
f(b)
0
(f,x 2 )
b
y,
8
=
O}
J f(x)dx = D). a For A = 0 we have for the differential equation M[y] = -y" = 0 on [a,b] so that ¢(.,o) and x(' ,0) are both linear functions on [a,b]; they are linearly independent since 0 is not an eigenvalue. {fEH y,o If(a)
f(b)
0
Consider now oH y ,8 e {¢o,x o }; let h be an element of S[h] E L(a,b); define g = S[h] which can then be regarded differential equation to determine h, i.e., i (ph)' + iph' solution of this equation which satisfies h(a) = h(b) = 0 (recall p(x) = x- 4 )
this space, then as a first-order regular = g on [a,b]. The can be seen to be
161
REGULAR DIFFI'RENrJAL EXPRESSIONS AND REL"THD OPER.1TORS
2 x 2 b 2 h(x) = ~i f t g(t)dt (x E [a,b]) with f t g(t)dt a a
o.
(4.90)
2 Since h' E L (a,b) this implies gr=L
For
f
2 (a,b).
(4.91 )
b
h(t}dt
0 to be satisfied we have
=
a
o=f
b
a i.e., from (4.90)
2 x 2 b3 b 2 b x5 x {f t g(t)dt}dx = :3 f x g(x)dx - f :3 g(x)dx a a a
0
(4.92 ) Since hE
H
o y,6
8 {¢o,x } we have to satisfy, see the proof of lemma 4.10(a),
o
f
b
S[h] '¢(. ,0)
f
0
=
a
b
S[h] i(·,O) = 0
a
it· ,0)
which, since '¢("O) and [a , b], imply
f
are linear functions and linearly independent on
b
g(x} dx =
f
b
xg(x) dx = O. a Hence if h is represented by (4.90) and h E oH ,6 8 {¢o,xo } then
(4.93)
a
y
and
f
b
xr g(x)dx
(r = 0,1,2,5).
0
=
(4.94)
a
Conversely if 9 satisfies the conditions (4.94) then h defined by (4.90) is an element of oH y ,6 8 {¢o,x o }' Now let fix) = flaY = fib} = 0 and
65 + f
a fix)
where a E C and So' Take h
E
H
o y,6
y
2
sx + yX + 6 (x E [a,b]) with a f 0 and let fix) dx = 0; it may then be seen that f is of the form
=
,6
a(x 5 + S x2 + y x + 6 )
(x E [a,b]) (4.95) o 0 0 are all fixed, non-zero numbers. Clearly f E H
o 0 8 {¢o,x } with 9 o (R h,f) o y,6
0 y,6
=
(CP(·,O;h),f) b
f a
=0
6
b =
f a
b
f
y,
M[CP(' ,O;h)] f
a
since f has the form (4.95).
S[h] satisfying (4.94); then
9f
S[h] f
WN. EVERITT
162
This last result implies ~
D(Q)
H
oy,6
8 {f} C
H
£
Oy,u
where {f} is the one-dimensional sub-space spanned by f of (4.95). This completes the examples. 5.
THE RIGHT-DEFINITE CASE
This section is devoted to stating the results for the right-definite case corresponding to those given in section 4 for the left-definite case. The detailed proofs for the results in this section follow closely those given in section 4, so that only appropriate reference is made to the relevant theorems and corollaries; the details are omitted. 5.1.
The right-definite boundary val ue problem.
For convenience the symmetric boundary value problem is restated; the differential equation is M[y] = AWy on [a,b] (5.1) or, equivalently, - (p(y' - ry))' - rp(y' - ry) + qy = AWY on [a ,b] (5.1 a) with boundary conditions, where
1 11, l] 211 ,
y, 6 E rL- 2
y[O](a) cos
y[l](a) sin
y
0
y[D](b) cos 0 - y[l](b) sin 0
0
y -
(5.2)
or, equivalently, yea) cos y(b) cos
y -
pry'
ry) (a) sin
y
0
pry'
ry)(b) sin
6
D.
(5.2a)
The coefficients p, q, rand w satisfy the basic conditions (2.1) and (2.7); the right-definite conditions (3.13), (3.14) and (3.15) hold throughout this section. Note that w is non-negative on [a,b], but can be zero or a subset of positive measure of [a,b]; see the remarks at the end of section 3(d). We work with the reduced Hilbert function space I;(a,b), see section 3(e), and denote the norm and inner-product in this space by 11.11 w and (.,.) w respectively. 5.2.
Pathological case.
There is no pathological case to consider in the right-definite problem, in comparison with the left-definite problem; see section 4.2. 5.3.
Eigenfunctions.
Lemma 4.3 holds for the right-definite case with the inner-product in (4.6) replaced by (~,~)w.
163
Rh'GULAR Dlf'f'ERENTIAL LX.J>RLSSIONS AND RELiTt']) OPERATORS
5.4.
Eigenva1ues.
Lemma 4.4 holds for the right-definite case; the corresponding proof is dependent upon the Green's formula (2.5) for the differential expression M. 5.5.
Orthogonal ity of the eigenfunctions.
Lemma 4.5 extends with the inner-product replaced by (. " )w; the result (4.14) is replaced by (m, n E N). (5.3) 5.6.
W has simple zeros. Lemma 4.6 extends to the right-definite case with (4.15) replaced by (5.4)
(A E C)
and all boundary conditions (5.2); the method of proof starts with (4.16) but recall that p is null on [a,b] in the right-definite case. As before this result implies that the zeros of Ware all simple (and real). 5.7.
The reso1vent function
The definition (4.26), with 1>: [a,b] x C definite case with b jJ(A) = exp[- J {r - rJ] a (i.e., jJ is independent of ,\) and
x
(f
E
Note that wf
E
S[ f]
wf
1wf 1}2
~ J
Lw2 • C, holds for the right(A
(5.5)
C)
E
L2). w
(5.6)
L(a,b) since
{j a
b
b
a
b
w
J w1f 12
<
00
a
from (3.14) and f E L2. w Lemma 4.7 continues to hold in the right-definite case with H
'(,6
5.8.
Properties of
replaced by
¢.
By and large the properties of ¢ are simpler in the right-definite case; mainly this is due to the fact all the right-definite problems considered here generate self-adjoint operators in the space l~(a,b); there is no equivalent to the symmetric, non-self-adjoint case for left-definite problems. Lemma 4.8(a) holds but always has the improved form given in theorem 5.9 (l ) (v) be 1ow. Lemma 4.8(b) is simplified; the residue of
¢
at the simple pole An is (x
E
[a,b])
(5.7)
W.N. EVHRITT
164
where k is again given in (4.22). n Lemma 4.8(c) holds without alteration; let the resolvent operator [2 -;. i 2 be defined for all A EO C {An: nEON} by
w
w
(R\ f) (x)
=
(x, A; f)
Lemma 4.8(d) holds but recall above.
(x EO [a, b] p
f EO
L~ (a , b) ) .
(5.8)
is null on [a,b] and S[f] is given by (5.6)
Lemmas 4.8(e) and (f) hold for the right-definite case without alteration. 5.9.
The self-adjoint case.
For all the right-definite problems the results given in theorem 4.9 continue to hold as in Theorem 5.9. Let the right-definite problem (5.1) and (5.2) satisfy the basic conditions (2.1) and (2.7); let the resolvent function be defined as in section 5.7 above; let the resolvent family {R } be defined by (5.8); then with H reA placed by [~ results (l)(i) to (v), and (2)(i) and (ii) hold as stated inY'o theorem 4.9. The operator T in the right-definite case is defined as in (4.48) and has all the same properties but now in 12, provided zero is not an eigenvalue which w can always be arranged on using the translation property given in section 3(f). -2
The self-adjoint operator T in Lw represents the right-definite boundary value problem (5.l) and (5.2).
-
Remarks. 1. We note that this theorem implies, as in theorem 4.9, that N = N = so that every right-definite problem (5.1) and (5.2), under the conditions stated, has an infinity of eigenvalues (all real and simple) and the entire (integral) function W(·) has an infinity of zeros (all real and simple). 00
2. The definition of the self-adjoint operator T gives an operator which coincides with the operator in L2 (a,b) generated by the expression w-1M, in the w case when w satisfies the additional condition w(x) > 0 (almost all x EO [a,b]); see again the remark at the end of section 3(d). The identification of the operators in the two definitions is given by showing, with w > 0 on [a,b], that {R } A -1 is the resolvent family of the operator generated by w M. Proof. The proof of theorem 5.9 follows the same 1 ines as for the proof of theorem 4.9; the details are omitted. 0 Corollaries 4.9(a) and 4.9(b) extend as given to the right-definite case replaced by L2 and i 2. 0 with Hand H y,o y,6 w W
RcGUL4R DlFFl'.RbVTIA I. l:XPIU:SS[ONS ,1ND J(cLUl:D OPIOR.-Jl'ORS
5.10.
165
The symmetric case.
As mentioned in section 5.8 there is no need to consider this case separately for the right-definite boundary value problems. 6.
GENERAL REMARKS
We have considered certain of the spectral properties of the second-order, linear, symmetric differential equation M[yJ = AS[y] on [a,b], with associated, separated boundary conditions. In the right-definite case (see section 5), and in the left-definite, selfadjoint case (see section 4) it is shown that the boundary value problem can be represented by a self-adjoint operator in a suitably chosen Hilbert function space. The methods employed are those of classical complex function theory and operator theoretic properties in Hilbert space theory. In the left-definite, non-self-adjoint case no satisfactory operator representation of the boundary value problem seems possible but the existence of an infinity of eigenvalues is obtained by an adaption of the classical methods of Titchmarsh. We list here a number of open problems: (i) What can be said of the order of W (the generalized Wronskian defined in (3.4)) as an integral (entire) function on the complex plane? (ii) What can be said of the zeros of Wwhen the general boundary value problem of section 3(b) is neither left-definite nor right-definite? (iii) Under what conditions on the coefficients of the differential equation can asymptotic expansions for large values of A be obtained; are such expansions uniformly valid on [a,bJ? (iv) In the left-definite, non-self-adjoint case of section 5 do the eigenfunctions form a complete orthonormal set in H, or Hy,u~? y, u (v) Does the analysis of Daho and Langer [6 and 7J extend to the leftdefinite case when one or both of the boundary condition parameters y and 0 lies in the negative interval [- ±rr,O]?
166
W.N. EVERITT
References [lJ
Akhiezer, N. I. and Glazman, I. M. Theory of linear operators in Hilbert space: Volume 1 (Pitman; London and Scottish Academic Press; Edinburgh, 1981; translated from the third Russian edition).
[2J
Atkinson, F. V., Everitt, W. N. and Ong, K. S., On the m-coefficient of Weyl for a differential equation with an indefinite weight function, Proc. London Math. Soc. (3) 29(1974), 368-384.
[3]
Coddington, E. A. and Levinson, N., Theory of ordinary differential equations (McGraw-Hill; New York, 1955).
[4]
Coddington, E. A. and deSnoo, H. S., Regular boundary value problems associated with pairs of ordinary differential operators, Proceedings of Equadiff 78 (International Conference on Ordinary Differential Equations and Functional Equations, Florence, Italy 1978).
[5]
Coddington, E. A. and deSnoo, H. S., Differential subspaces associated with pairs of ordinary differential expressions, J. of Diff. Equations 35 (1980), 129-182.
[6J
Daho, K. and Langer, H., Some remarks on a paper by W. N. Everitt, Royal Soc. of Edinburgh (A) 78 (1977), 71-79.
[7]
Daho, K. and Langer, H., Sturm-Liouville operators with an indefinite weight function, Proc. Royal Soc. of Edinburgh 78(1977), l61-19l.
[8]
Dunford, N. and Schwartz, J. T., Linear operators: Part II (Interscience; New York, 1966).
[9J
Eastham, M. S. P., Theory of ordinary differential equations (Van Nostrand Reinhold; London, 1970).
Proc.
[lOJ Everitt, W. N., Some remarks on a differential expression with an indefinite weight function, Mathematical Studies 13 (1974), 13-28 (North Holland; Amsterdam, 1974; edited by E. M. de Jaeger). [11J Everitt, W. N., An integral inequality associated with an application to ordinary differential operators, Proc. Royal Soc. Edinburgh (A) 80 (1978), 35-44. [12J Everitt, W. N., On the transformation theory of ordinary second-order linear symmetric differential equations, Report DE 81: 1, Department of Nathematics, University of Dundee. [13] Everitt, W. N. and Race, David, On necessary and sufficient conditions for the existence of Caratheodory solutions of ordinary differential equations, Quaestiones Mathematicae 2 (1978), 507-512. [14J Everitt, W. N. and Zettl, Anton, Generalized symmetric ordinary differential expressions I: the general theory. Nieuw Archief voor Wiskunde (3) XXVII (1979),363-397. [15J Kamke, E., Differentialgleichungen: Losungsmethoden und Losungen (Chelsea; New York, 1971; reprinted from the third edition, Leipzig, 1944). [16J Naimark, 1968) .
M. A., Linear differential operators: Part II, (Ungar; New York,
REGULAR DIFFERENTiAL EXPRnSS]ONS AND RFL1TrD OPERATORS
167
[17J Niessen, H. D. and Schneider, A., Spectral theory for left-definite systems of differential equations: I and II, Mathematical Studies 13 (1974), 29-44 and 45-56 (North Holland; Amsterdam, 1974; edited by E. [q. de Jaeger). [18J Pleijel, Ake, A positive symmetric ordinary differential operator combined with one of lower order, Mathematical Studies 13 (1974),1-12 (North Holland; Amsterdam, 1974; edited by E. M. de Jaeger). [19J Pleijel, Ake, Generalized Weyl circles, Lecture Notes in Mathematics 415 (1974), 211-226 (Springer-Verlag; Heidelberg, 1974; edited by 1. M. Michael and B. D. Sleeman.) [20J Riesz, F. and Sz.-Nagy, B., Functional analysis (Ungar; New York, 1955). [21] Taylor, A. E., Introduction to functional analysis (Wiley; New York, 1958). [22J Titchmarsh, E. C., Eigenfunction expansions I (Oxford University Press, 1962)
This page intentionally left blank
This Page Intentionally Left Blank
Spectral Theory of Differential Operators I. W. Knowles and R. T. Lewis leds.) © North-Holland Publishing Company, 1981
AN EIGENFUNCTION EXPANSION ASSOCIATED WITH A TWO-PARAMETER SYSTEM OF DIFFERENTIAL EQUATIONS Melvin Faierman Department of Mathematics University of the Witwatersrand Johannesburg South Africa Dedicated to Professor F.V. Atkinson on his 65th birthday
In this work techniques from the theory of partial differential equations are used to prove the uniform convergence of the eigenfunction expansion associated with a left definite twoparameter system of ordinary differential equations. INTRODUCTION We consider the simultaneous two-parameter systems (p] ( x] )y; ) + (A] A] (x] ) - A2 B] (x] ) - q] (x 1 ) ) y] 0, 0 <; x] <; 1, I = d/dx , (1 ) (2) y](O)=y](l)=O, and (p2(x2)y~)'+(-A] A2 (x 2 )+12 B2(X2)-q2(X 2 ))Y2=0, 0<;x 2 <;1, =d/dx ,(3) Y2(0) =y (1) =0, (4) where the ~i' qi' Ai' and Bi are real-valued, sufficiently smooth functions with Pi> 0, qi - O. I
I
Writing \ for (A] '\2)' we call A* an eigenval ue of the system (1-4) if for A= \*, (2r-l) has a non-trivial solution, say Yr(xr ,\*), satisfying (2r) for r = 1,2. The product ydx] ,\*) Y2 (X2 ,\*) is called an eigenfunction of (1-4) corresponding to A* . Then under the assumption that II = A] B. - A2B] * 0 in 12 (the product of the intervals 0 <; xr <; 1, r = 1,2), Faierman t1978) has established a sufficient condition for a function defined on I2 to be expanded here in a uniformly convergent series involving the eigenfunctions of the system (1-4). However, it is as pointed out by Atkinson (in a private communication) that in many practical applications the hypothesis II + 0 in 12 may fail to hold, and hence in this work we shall be concerned with the eigenfunction expansion associated with the system (1-4) under the hypotheses that: (i) II assumes both positive and negative values in 12 ,and (ii) there exist real numbers T] ,T 2 such that I] Ai + T2 Bi > 0 in 0 <; xi <; 1 for i = 1,2. On account of (ii), we may henceforth suppose that Ai > 0 in 0 <; xi <; 1 for i = 1,2. Writing x for (x] ,x 2) and denoting by ¢r(xr ,A) the solution of (2r-l) satisfying q,dO,A) = 0, Pr(O)¢!-(O,A) = 1, r = 1,2, let 1jJ*(x,\) = 'h(x] ,A)q,2(X2 ,A). Let (l denote the interior of 12 . Then our first result asserts that THEOREM 1. The totality of the eigenvalues of the system (1-4) forms a countably infinite subset of E2 having no finite points of accumulation. Moreover, if At and \" are any two distinct eigenvalues of (1-4), then J(l lI(X) 1jJ*(x,At)1jJ*(x,\~)dx = 0 and J(l lI(x)(1jJ*(x,\t))2dx , 0 Next let {An}, n ~ 1, denote an arbitrary, but fixed enumeration of the eigenva7ues of (7-4). For nzl, let 169
M. FAIERMAN
170
~n(x)
and for
= ~*(x,An)/lf0
1
6(x)(~*(x,An))2dxI2
,
f, g ( L2 (0), let
(f,g)ll = Then we observe from theorem 1 that (~n , ~n) 6 = P n =
* 1.
I0
6fg dx.
(~m '~n)ll
0
if m
* n,
while
The following theorem contains the main result of this work. In this theorem Xl = (xix E 12 ,6(X) = O}. We shall also say that a series of functions all defined on a given set X converges regularly on X if the series of absolute values of these functions converges uniformly on X . THEOREM 2. Let f(x) be a function of class C4 on 12 which vanishes in a relatively open subset of 12 containing X" Let f(x) and its. partial derivatives up to and including the second order vanish on f= 12 - Q. Then f(x) = I Pn(f'~n)ll ~n(X) + h(x) for x ( 12, no> 1
where the series converges regularly, nnd hence uniformly, on 12 ' and h(x) denotes a continuous function on 12 satisfying ll(X) h(x) = 0 for x c 12 . Finally, if Xl is a set of measure zero, then h(x):= 0 and the above series converges uniformly to f(x) on 12 PRELIMINARIES Associated with the system (1-4) is the boundary value problem (5 ) Lu - A6(X)U = 0 for x E 0, u(x) = 0 for x E r, where L denotes the elliptic operator 2 -[ I Dr ar(x)D r - q(x)J, r=l D = a/ax , a l = Pl A2 ' a 2 = P2 A, , and q = q] A2 + q2 A]. In order to deal w~th (5) ~e firstly fix our attention upon the boundary value problem Lu = f for x E 0, u(x) = 0 for x. r, (6) for f E L2 (Q). To deal with this problem we introduce the space V which is the completion of C~(0) with respect to the norm U Ul 0 ,where we refer to Agmon (1965) for terminology. It is clear that V is a dosed subspace of H, (,,) and that an element u in Hl (0) belongs to V if and only if the trace of u on f is zero. On V we define the form 2
B(v,u) =
I
(Drv, arDru)+(v,qu), r=l where (,) and U U, without subscripts, denote the inner product and norm, respectively, in L2(1l). It is clear that B(u.u);> 0, IB(v,u) I <: kUvU] 0UuU 1 Q for some constant k, and that B(v,u) is coercive on V. We also assert th~t B(v,u) is strictly coercive on V; i.e. B(u,u) ;> k UuU~ Il for some positive constant k. To see this, we argue as in Mizohata (1973) and Faierman (1978) to show that B(u,u) = implies u = O. Hence if A is the selfadjoint operator associated with B, then A;> 0 and the nullity of A is zero. Thus for A > 0, A + AI is certainly Fredholm and has index zero, while a simple argument shows that -AI is relatively compact with respect to A + AI. It now follows that zero is in the resolvent set of A, so that B(u,u);> kUull z for u E D(A) = domain of A, and hence for u E V, since D(A) is a core of B (here k denotes a positive constant). The assertion now follows from this result and the fact that B(u,u) ;> k][UD l uI1 2 + UD 2uU 2] for some positve constant kl .
°
We next consider the generalized boundary value problem: given a u c V such that
f
E
L2 (0),
find
171
AN liIGFNFUNC110N EXPANSION
(7) B(v,u) = (v,f) for every v (V. From Faierman (1978) we know that (7) has a unique solution u. Moreover, (i) u ( H~OC(~), u (H 2(G) for every open semi-disc G with edge on r and whose closure does not contain any corner points of r ,and u is both a strong and weak solution of (6); (ii) u is continuous in any compact subset of 12 which excludes the corner points of rand u = 0 at each point of f which is not a corner point; (iii) if f f H2(G) for every open disc G contained in ~ and for every open semi-disc of t~e kind described above, then u is of class [2 in any compact subset of 12 WhlCh eXCludes the corner points of f.
If u denotes the solution of (7), then let us introduce the notation u = Tf. Then T is a bounded linear transformation of L2(~) into V , and hence it follows that as a mapping of L2 (n) into itself, T is positive, compact, and has nullity zero. Moreover, from Faierman (1978) we know that the range of T, R(T), is lulu F V, U e HIOC(n), Lu • L2(~)}' We now introduce the operator A in L2 (n) by defining D(A) = R(T) and putting Au = Lu for u e D(A). Then A is selfadjoint, A and T are inverses, and moreover, this operator A is identical to the selfadjoint operator A associated with the form B(v,u) introduced above. THE BOUNDARY VALUE PROBLEM (5) We are now going to use the above results to derive some information concerning the boundary value problem (5). To this end we let Q denote the bounded, selfadjoint operator on L2(~) defined by (Qf)(x) = ll(x)f(x). If K = TQ, then K is a bounded 1 inear transformation of L2 (~) into V , and hence as a mapping of L2(~) into itself, K is compact. Moreover, K is strongly symmetrisable with respect to Q , and is also symmetrisable with respect to the compact, positive operator S = QK. Hence from Zaanen (1953) we know that the characteristic values of K are all real, each of finite multiplicity, and at least finite and at most denumerably infinite in number. Moreover, if we denote the characteristic values by {lJn}, n 0: 1 , arranged in increasing order of magnitude and with each being counted as often as its multiplicity indicates, then 0 < l\lll <: 1\l2l <: .,. and IPnl ->as n ->if there are infinitely many characteristic values. Finally, to the sequence of characteristic values {lJn} there corresponds the sequence of characteristic functions fUn}' n ~ 1, where (Sun ,urn) = l\lnl- 1 if m = nand is zero otherwise, (Qu n ,urn) = sgn \In if m = n and is zero otherwise. 00
00
Let us call the complex number P a Q-eigenvalue of A if there is a non-zero element u E D(A) such that Au = \lQu; u is called a Q-eigenfunction of A corresponding to \l. The set consisting of all Q-eigenfunctions corresponding to \l together with the zero element of D(A) forms a subspace of L2(~) whose dimension will be called the multiplicity of the Q-eigenvalue v. It is now easy to see that the Q-eigenvalues of A are precisely the characteristic values of K, {)In}, and the corresponding Q-eigenfunctions are precisely the characteristic functions of K, {un} Finally, we note from the above results and the fact that un = \lnKun = \lnTQun ' that for each n, un is of class [2 in any compact subset of 12 which excludes the corner points of r and that un = 0 at each point of r which is not a corner point. PROOF OF THEOREMS 1 AND 2 If A* =(Af ,A~) is an eigenvalue of the system (1-4), then we say that real if both Af and A; are real. Faierman (1979) has shown that the (1-4) possesses real eigenvalues which form a denumerably infinite subset having no finite points of accumulation. We shall use this fact to prove Accordingly, it is a simple matter to show that if At =(At ,A!) and A# two distinct eigenvalues of (1-4), then J~ ll(X) ~*(x,\t) ~*(x,\#)dx = O.
\* is system of E2 theorem 1. are any It is
M. FAIERMAN
172
also easy to show that ¢(x) W*(X,A t ) E D(A) and A¢ = A~ Q¢. Thus A! is a Q-eigenvalue of A and so it is real and * O. Hence the eigenvalues of (1-4) are all real. Also, since ¢ = A! K¢ , it follows that (A!)-l(Q¢,¢) > O. This completes the proof of theorem 1, and moreover, we have also shown that each eigenfunction of (1-4) is a characteristic function of K corresponding to some characteristic value. On the other hand, we may employ the arguments of Hilbert (1953) to show that the characteristic functions of K may be chosen so that the sequence {un}' n ~ 1, is but a rearrangement of the sequence of eigenfunctions of the system (1-4), {wn}, n 2 1. We next assert that (8) I u~(X)/lllnI3 -:; C for n21 where C denotes a constant independent of x. To prove this assertion, we argue as in Faierman (1978), making use of the facts that: (i) ~~ TS un = un for n 2 1 and (ii) shs~ is a compact positive operator with characteristic values {~~}, n ~ 1, and corresponding to this seqyen~e of characteristic values is the sequence of characteristic functions {1~nIZ Szu n}, n ~ 1, which forms an orthonormal sequence in L2 ((J) .
Turning to the proof of theorem 2, it is clear that f, as defined in the theorem, is in D(A). Next let gl (x~ (6(X))-1(Lf)(x) for x E I2 - Xl ,gl (x) = 0 for x E Xl' Then clearly gl E C in I~ and vanishes on r, and hence it follows that g] E D(A). Moreover, if g(X)=\6(X))-1(Lg])(x) for x E I -Xl ,g(x) = 0 for x E Xl ' then it is clear that ~ is continuous in I, . T~us in L2 ((J), Af = Qgl ' Ag] = Qg, and hence f = K g, from which it follows that f = L sgn ~n(Qf,un)un + h , (9) n?l where Qh = 0 (Zaanen (1953)). On the other hand it is easy to see that for each n , 1
1
1
3
sgn ~n(Qf,un)un(x) = (SZg, l~nl2 S2 un)un(x)/I~nlz for x E 12 ' and hence it foIlow~ from (8) and the above remarks concerning the characteristic functions of S2 TS2 , that the series in (9) converges regularly on I2 . The assertions of theorem 2 follows from this and the preceding results. REFERENCES [lJ Agmon, S., Lectures on Elliptic Boundary Value Problems (Van Nostrand, New York, 1965) . [2J Faierman, M., Eigenfunction expansions associated with a two-parameter system of differential equations, Proc. Roy. Soc. Edinburgh 81A (1978) 79-93. [3J Faierman, t~., An oscillation theorem for a two-parameter system of differential equations, Quaestiones Math. 3 (1979) 313-321. [4J Hilbert, D., GrundzUge einer Allgemeiner Theorie der Linaeren Integralgleichungen (Chelsea, New York, 1953). [5J t1izohata, S., The Theory of Partial Differential Equations (University Press, Cambri dge, 1973). [6J Zaanen, A.C., Linear Analysis (North-Holand, Amsterdam, 1953).
Spectral Theory of Differential Operators I.W. Knowles and R. T. Lewis leds.) © North-Holland Publishing Company, 1981
.. DISTRIBUTION OF THE EIGENVALUES OF OPERATORS OF SCHRODINGER TYPE J. Fleckinger
U. E. R. Maths. Univ. P. Sabatier 31062 TOULOUSE - FRANCE
We obtain an asymptotic estimate for the number of eigenvalues less than s for an operator of Shrodinger type: Aq = A + q defined 0" an unbounded domain in IR n when A has unbounded coeffi ci ents. I - INTRODUCTION It is well known that the spectrum of Hq = -6 + q defined on IR n is discrete when the potential q is a positive smooth function, tending to +m at infinity. N(s,Hq,IR n ), the number of eigenvalues of Hq less than s, is such that: N(s,H ,R n ) '" c f (s-q(x)) n-2 dx q {XEIR n !q(x)<s}
s -). +00
We generalize here this formula and obtain an estimate for an operator of Schrodinger type: A = A + q defined on an unbounded domain It c lRn, with zero q Dirichlet boundary conditions; A is formally selfadjoint, ell iptic of order 2m, with smooth coefficients tending to +00 at infinity. With suitable hypothesis on A, q and It, we prove the following: (0)
S
-r
+00
where Its
{xEltjq(x)<s) lJ(x) = (2rrr n meas{t;E IR"jA'(x,s) <
n
A'(x,!;) is the symbol of the leading part ofA. For example we obtain the asymptotic distribution of the eigenvalues of:
When k
=
0, L is the usual Schrodinger operator.
Many results are known when A, defined on IRn, has constant coefficients [3,5]. When It is the outside of a bounded domain, D. Robert [4] proves (0). 173
]. FLFCKINGER
174
In [ZJ we obtain (0) when [J is an unbounded domain in lRn, and A has bounded coefficients. To prove this result, we use the "max-min" principle [1] and some consequences: partition of the domain; comparison of A with a homogeneous operator with constant coefficients. II - HYPOTHESES AND RESULTS
Let m be a positive integer and
Q
be an unbounded domain in lRn.
(1) Let p and q be two continuous functions defined on Q, real-valued bounded below by 1, which can tend to +00 at infinity; we suppose that:
p-Zm(x) q(x) ~ +00
when
Ixl ~ +00.
Let us denote by V~(Q) the completion of C~(Q) with respect to the norm II I q, Q where: I uI
{J [ I
=
Q
q,
p
labn
Q
Ia I (x) Ioau (x) 12 + q (x) Iu (x) IZJ dx} liZ
a = (al,···,a n ) E INn and Oa is a derivative of order lal = a l + ... + an. V~(Q) is a Hilbert space and it is simple to verify that:
~BQ~Q~lI1Q~_1:
The imbedding V~ into LZ(Q) is compact.
Proof. We use the classical criteria of compactness for unbounded domains and the following inequality: I ul Z2
L (Q')
~ sup q-l(x) Q'
fQ
q(x)lu(x)I
Z
dx
~
E(R)
IIull~,QR'
R R R where QR = {x E [J / Ixl > R} we notice that E(R) tends to 0 when R + +00. Let aq be an integrodifferential form, continuous and coercive on V~([J):
(2)
a (u,v) = (a+q)(u,v)
=
q
f ( T Q
I a T.::.m
a sex) Oau(x) OSv(x) + q(x)u(x)VTXTT dx a
I si.::m for (u,v) (3)
E
VO((l) q
x
VO([J).
We suppose that:
q
a
as
=
-a-
Sa
E
CO(i'l) and that:
Let us denote by AO the positive selfadjoint operator, unbounded in q associated by the Lax-Milgram theorem, to the variational problem (V~([J), L2([J), aq).
2
L (n),
175
DISTRIHUTION OF UGh",!' lLUES OF SCHROEDINGJ:R OPhRATORS
We deduce from the Proposition 1 that AO has a discrete spectrum consisting q of isolated eigenvalues: s.
---r
J j
-+
+00
+00
(each eigenvalue is repeated according multiplicity). We study the asymptotics of the number of eigenvalues less than s: N(s,A o ,Q) = N(s,Vo(Q),a ) = card{j E JIl / sJ. < s} when s q q q We give now some assumptions concerning A and Q.
->
+00.
q
There exist two positive numbers EO and s· such that be extended to ~ = {x ERn / dist(x,Q) < EO} and (4)
\I EEl 0, E [, \Is > s· o \I(x,y) E
Qx n
\Ill .':. lls q(x)
=
<
p,
q and a
ex8
can
",sup (p(x) q-l/2m(x)) 1/2 {XEQ/ q ( x ) > S } s, q(y)
<
s, Ix-yl
<
In
Tl
=>
Ip(x) - p(y)1 .':. Ep(X) 1q(x) - q(y) 1 .':. Eq(X) laexs(x) - aas(Y) 1 ~ Elaexs(x) I. When w is a subset of ~, we denote by Vl(w) the set of the restrictions to w of functions in VO(Q) and by Aql the realiz~tion of the variational problem 1 2 q (vq(w), L (w), aql. (5)
We suppose that a is uniformly coercive, i.e.:
For any positive number s, ns set and (6)
=
f
L pex (x) 1Oexu (x) 12 dx. w lal~m {XE$,/q(X) < s} is a Lebesque measurable
a (u , u1 .:: y 2
where ens] = f
p-n/2 (xl dx. liS
n We consider a partition of R into non overlapping cubes (Q~l~EZn with side and centers xI:; and we suppose that L 0 (n 2/p )n/2
(7) 11
I:;EI \1
1;
--->11 -+ 0
0
\Is > s"
176
]. HECKfNGER
We prove here the following result: THEOREM 1: ---------
We suppose that the hypotheses (1) to (7) are satisfied; then (0) holds, i.e.:
N(s,A~,Q) ~
~(x)(s_q(x))n/2m dx
f
S
--r
+=
Q
s
with
Qs )l(x)
{x
Q/q(x) < s} (21fr n meas{i; E JRn/A' (x,d E
<
l).
Remarks: When Aq = -A + q and n = JRn, we find again the usual formula. When n is bounded, q(x) is bounded, and Qs = n for s big enough; we obtain again the well known formula for elliptic operators on bounded domains (with suitable hypothesis) [1,4]: N(s,Ao,n) ~ N(s,Ao,n) ~ sn/2m q
(8)
When lim [n] is finite, that means S++oo s again that (8) holds.
fQ
f
p-n/2(x) dx
)l(x) dx
s
+
+00.
Q =
f n )l(x)
dx
< 00,
we find
I I I - THE "MAX-MIN" PRINCIPLE We shall recall briefly some well known results on the "max-min" formula and some consequences. Let (V,H,a) be a variational problem where, as usual, H is a Hilbert space; V is a subspace of H such that the imbedding of V into H is compact with dense range; a is a hermitian form, continuous and coercive on V. By Lax Milgram's theorem, we associate with this problem an operator A which is positive, selfadjoint, and unbounded in H. We deduce from the compactness that the spectrum of A is discrete; the eigenvalues Sj are given by the "max-min" formula 1: (9)
where 9j is the set of all j dimensional linear subs paces of H. Let us denote as above: N(s,V,a) = card{j E ~ / Sj ~ s}, the number of eigenvalues of A less than s. We deduce from (9) the following results: E~QEQ~!I!Q~_~:
3E then:
>
Let (V,H,a) and (V,H,b) be two variational problems such that: 0, 3c
>
0, Ifu
E
V Ia ( u , u) - b (u , u) I ~ Ea (u , u) +
q uII ~
177
DISTRIlJUTION OF EIGENV IILt'ES OF SCHROEDINGfiR OPERATORS
N[O-ds - C, V, b) :5- N(s, V, a) :5- N(s(l+d + c, V, b). We suppose now that Q is an open set in R n and Hm(Q) and Hm(Q) are the o usual Sobolev spaces on Q. Let a be an integrodifferential form, hermitian, continuous and coercive on Hm(Q). We denote by Al Crespo AO] the realization of the variational problem (Hm(Q), L2(,,), a) Crespo (H~("), L2(Q) ,a)]. ~~Q~Q~~!~Q~_~:
Suppose that Ql and Q2 are two disjoint open sets in Rnsuch that: = "1 U Q2; then, the following holds: l l N(s,Ao ,Ql) + N(s,Ao '''2) :5- N(s,Ao ,Q) :5- N(s,A ,,,) :5- N(s,A ,Ql) + N(s,A l '''2)' where N(s,Ao,n) = N(S,H~(,,),a) and N(s,A l ",) = N(s,Hm(Q),a). Remark: This result can extend to other spaces and, in particular, proposition 4 holds for the spaces V~(Q) introduced above (i = 0 or 1 correspond to different boundary conditions). IV - A FIRST ESTIMATE Let us write:
f(s,q)
=
f
\l(x) (s_q(x))n/2m dx; Q
s
the following estimate holds. THEOREM 2:
Proof:
There exist two positive numbers c' and c" such that:
c' s n/2m [Q s J -< f(s,q) c " S n/2m [ Q s ] We deduce from the coerciveness of aq that:
lat=m
\;Js .:: s" •
r;2a
hence: \l(x) :5- cp-n/2(x), and we obtain the upper bound. To obtain the lower bound, we use hypotheses (3) and (6). We can write: f(s,q) .::
f
\l(x) (s_Q(x))n/2m dx .:: c[Qs] sn/2m.
ns/ 2 V - ESTIMATES FOR AN OPERATOR WITH CONSTANT COEFFICIENTS ON A CUBE Let A~ be the operator defined on Q~ associated with the hermitian form: (10)
a (u,v) I;
=
f Q
1
\aS <m <m
aaB(X~) Oau(x) OSv(x) dx ;
I; we denote by A~ the leading part of AI; associated with the form following results: ~~gEg~!I!g~_~:
a~.
There exist two positive numbers 60 and Y4 such that:
We have the
178
]. FLECKINGER
'tI Ii
'tI s .:: S", 'tI ~ E I, 'tI n .2 11S' ::'IT t; = <5 1- 2m +
.:: Ii 0'
N((1-o)s-Y4'~' A~i, 0sh N(s, A~, Ot;) P~QPQ~EIQ~_§:
There exi s ts Y5
0) _ IN(s , A,i r;' 1; where
)1r;
=
)11;
>
<5 -
1 (p 1;/ 11 2) m- 1 :
.2 N((1+0)s+Y4 't;'
0 such that:
A~i, 01;)'
'tis.:: s", 'tIr; E I, 'tin .2 11S:
10 1 sn/ 2m l < s(n-l)/2m (2/ )(n-l)/2 1; - Y5 11 Pr;
ll(Xr;)'
Proof of Proposition 4: We use the interpolation inequalities; hence: for all a small enough, for all u E V~(~), for all integers p and q less than m and such that p + q < 2m: (u) (u) < a(u)2 + c(a l - 2m + a- l (p /n2)m-l with p q m t;
(u)~ = f
(11 )
pp r;
Or; Then, by (5) : 300
loau(x) 12 dx where C does not depend on
3Y4
>
r;.
0,
0,
'tI0 -< 00'
'tI~
E I: 2
lar;(u ,u) - a~(u,u) I ~Ear;(u,u) + Y4'd! ul 2 L (0)
(12 ) with
>
I
lal=p
'r;
defined as above, and we use proposition 4.
Proof of Propos it i on 5:
Let us denote:
B = (_l)m I oa (0 -m a (x) OB). 1; 1aT =m r; as 1; i -m i 2m -m i Pr;' Br;' 00 ) N(s, A~ , 01;) = N(sPr; , Br;' Or;) = N(sn where 00 is the homothetic cube of Or; with center xr; and side 1. It is well known [4J that: 3 Y5 > 0 'tis > 0 IN(s, B~, 0 ) - llr;(B) sn/ 2m l ~ Y5 s(n-l )/2m 0 where )11;(B) = meas{~ ERn / B' (~)
<
l}
VI - PROOF OF THEOREM 1 Let s > SOl and E ~ EO be two positive given numbers. (13)
I
r;EI
because 'tIw c {x/q(x)
By proposition 5:
l O 0 N(s,A ,0 ) < N(s,Ao,Q) < I N(S,A q ,Or) q r; q - r;E I S
>
1 s}, N(s, Aq,w) = O.
We take n small enough so that, by (4), on each cube 01;: (14)
laq(u,u) - (a 1; + q1; )(u,u) 1 -< €a q (u,u). We use propositions 3 and 6 and (11), (1) and (14):
DISTRIRUTION OF GGbV! '.4 U TS Of' SCHROEDINGER Orr;RATORS
where TZ; is given by (12) and s~ = (1
±
d
s -
q~
with qt; = q(Xt;)'
-2 Let us choose o = s-l/d with d = 2m - 1.2 , and 11 such that sl/2m = Ps n We have: r l (N(s,Ao,0) - F+):"f- l . L IN((1+6)S++1'4 T ,A,l, Q )-jJ 10 1 ((1+o)s++1'4T )n/2ml q Z;EI Z; Z; Z; Z; Z; Z; Z; Z; + f-l.
L jJz;Qz;[((1+6) s; + Y4Tz;)n/zm - sn/2m J ,
Z;EI where F+ F- =
=
L
z;EI
((1+6)5+ + 1'4 T )n/2m jJ IQ I, Z; Z; Z; z;
L ((1-6)S- - Y4
Z;EI
Z;
)n/2m IQ
T
Z;
1
Z;
and f- l
\l
l/f(s,q).
Z;
By letting s + +00, € + 0, we have the upper bound. bound by an analogous calculation.
We obtain the lower
[1]
REFERENCES Courant, R. and Hilbert, D., Methods of mathematical physics, Interscience.
[2J
Fleckinger, J., Note au Cras, Paris, Serie A, t.
[3J
Reed, M. and Simon, B., Mathematical Physics, Academic Press.
[4]
Robert, These Universite de Nantes (1977).
[5]
Titchmarsh, E., Eigenfunction expansions, Oxford.
179
This page intentionally left blank
This Page Intentionally Left Blank
Spectral Theory of Differential Operators I.W, Knowles and R. T. Lewis (eds.) © North-Holland Publishing Company, 1981
THE LCX::AL ASYMPTOTICS OF mNTINUUM EIGENFUNCrION EXPANSIONS
S. A. Fulling Mathematics Depart:rrent Texas A&M Universi ty .jCollege Station, Texas U.S.A.
and Institute for Theoretical Physics University of California Santa Barbara, california U.S.A.
'TWo theses are advanced: (1) The study of "spectral invariants" can and should be extended to operators with continuous spectra. (2) The subject is closely related to the asymptotic approximation of eigenfunctions by a local amplitude and a phase integral. This program has been carried out in the case of vector-valued functions of one variable. It is well known that the various integral kernels, or Green functions, associated with a self-adjoint differential operator have asymptotic expansions at short distances, in which the coefficients are geometrical invariants constructed in a local manner from the coefficient functions of the operator [18,10,17,19,20,9,13]. When the spectrum is discrete, by integrating certain of these quantities over the domain or its boundary, one obtains infoi:nation about the asymptotic distribution of the eigenvalues [7,2] and about the global structure of the domain region or manifold itself [26,23]. But the local quanti ties thernselves do not depend on whether the spectrum is discrete or the dornain compact. Also, the local objects obviously contain more detailed spectral information than their integrals do. The expansion coefficients for various kernels are related to each other in simple ways, and all of them must stem ultimately from a local asymptotic representation of the spectral projections of the operator [19,20,9]. Nevertheless, the bulk of the literature concentrates on compact dornains and discrete spectra, and in recent years there has been surprisingly little work on carrying back results about Green functions to obtain information about the spectral decomposition. Today I am reporting the first step in an attempt to develop a unified approach to this subject, emphasizing the central role of the spectral decomposition 1 or continuum eigenfunction expansion. The main point I shall emphasize is that considerable information about the spectral projections can be obtained directly (not from a Green function) by means of a suitably sophisticated version of an old favorite tool of the physicist: the WKB approximation for the individual eigenfunctions. Incidentally, this project was motivated by a very practical need for more information about the eigenfunction expansion in carrying out calculations in quantum field theory - I 'll return to that later. So far I am prepared to speak only about the relatively trivial case of a single independent variable, but I can handle any nurrl:>er of dependent variables. So, let M be one of the four possible one-dimensional manifolds (the circle, the interval, the half line, and the whole real line) , and consider functions on M whose values are vectors with r complex components: a -j-
Permanent address 181
=
I, . ", r.
STEPHEN FULLING
182
In fancier language, ¢ is a section of a vector bundle. K= - d
2
X
lJet
+ Vex) ,
where vex) is an Hermitian rratrix. (A much larger class of operators can be put into this norrral form by change of variables [15,16]. In the case of the circle, one point may have to be left out.) Impose boundary conditions sufficient to make K self-adjoint. (The circle is treated as an interval with boundary conditions relating the two ends.) For simplicity, assurre that K is positive definite and V is sm:::>Oth. In one dimension the spectral theorem for K is expressed very explicitly by the Titchmarsh-Kodaira eigenfunction expansion theory [28,21,22,27]. For simplicity I review this for scalar functions only, but the formulas apply to the vector case when the symbols are reinterpreted as vectors and matrices. Choose a point x € M. The eigenfunctions ljJ,j (A € j = 0 or 1) are the classical solutions of 4e differential equation Kl~Aj = AljJ Aj with initial data
*,
In general they will not be square-integrable nor satisfy the boundary conditions. There is an analogue of the Fourier transform: fk(A) =: JMljJAk(x) f(x) dx, f(x)
fa
1 'k L: ljJA' (x) dj)J (A;X O) fk(A) j ,k=O J
jk where the fl are certain Stieltjes measures with support in the spectrum of K. The functional calculus is F(K) =:
J'"0
F(A) dE A '
from which it follows that the integral kernel of the spectral projection EA is E, (x,y) A
= JA
'k
L: ljJ ,(x) dj)J (0) ljJ key) 0 ' k OJ 0 J,
On the diagonal, this and its derivatives reduce to the spectral measures:
Notice that this forrralism is tailor-made for studying the behavior of things at xo, even when the spectrum is discrete so that, traditionally, norrralized eigenfunctions would be used instead. The heat kernel of K is H(t,x,y) (The solution of dtU we have
OO
= J0
e
-At
- Ku(t,x) with u(O,x)
dE (x,y) A
=
f(x) is u
= Hf.)
On the diagonal
H(t,x o ,xc) JxH(t,x o ,xc)
e
-At
dj)
10
(A;X O) ' etc.
The derivatives of H have not often been studied, but they are needed to obtain the complete local spectral inforrration about K. In fact, on higher-dimensional
LOCAL ASYMPTOnCS
or C()NTTNUUM EIGENFUNC'I10N LXPANSIONS
manifolds we will need derivatives of arbitrarily high order (121. only those of orders 0 and 1 in each variable.
183
Here we need
It is known [17,15,16,301 that as t t 0, H and its x and y derivatives at Xo have asymptotic expansions (which also may be differentiated term by term in t); for example, d d R(t x
x y
,
x) 0'
'c
(4rr)-l/2
0
'z
v=O
Ell (x ) t v -(3/ 2 ) . v 0
Each E~ 1 (x o) is a polynomial in V(x o ) , V' (x o)' V" (x o )' ••• , in each term of which: the sum of the orders of the derivatives, plus twice the nurrber of factors, equals 2v. In other words, V and E-(, 1 can be regarded as having the dimensi0r:is of [lengthr 2 and [length1- 2v , respectively. (Remember that V, H, and can all be r x r matrices.)
Et
These asymptotic series for the heat kernel at small t are related to the asymptotic behavior of the spectral measures at large A, but in a subtle way. It is easy to see that if there exists an expansion dllll(A;X ) o
"u
l IT
l:
v=O
pll(XC) ,}-2v dw , V
'
then the cited expansion for d d H forces xy nll _(2v-l)!!E ll 'v+l (_ 2) v v+l
ifv>O
(and no odd or fractional pc:wers of w can be present). Thus the coefficients in the expansion of dill l/dw are uniquely determined by the heat kernel. A similar analysis can be based on the expansion of the kernel of (K - z)-l as z ~ - 00 , or on various other kernels associated with K. The trouble is that in general this series is not asymptotic to dllll/dw - in fact, if K has any point spectrum, dllll/dw is rot even a function! If we integrate to get an expansion for the function Illl(A;X o) defining the Stieltjes measure, then the first term is a valid asymptotic approximation:
(by Karamata's Tauberian theorem [71 - see also [201); but the error term here can be a zigzag function Ivith jumps as large as u/ for arbitrarily large w, so there is no "next term" in an expansion of Illl in pc:wers of w. Nevertheless, series like this have been given precise mathematical significance in terms of various averaging procedures [3,1,91, of which perhaps the best is to relate them to genuinely asymptotic approximations to the iterated indefinite integrals, or equivalently the Riesz means, of the quantities being expanded [24,19,201. But the real, practical significance of these series is that, as I mentioned at the beginning, there is a whole family of quantities whose singular behavior in some lifnit is dictated by them. This includes, besides the heat kernel, the kernel of (K - z) -1, the zeta function (kernel of K""""S), the kernel of exp(- tKl/2 ) (which solves the Dirichlet problem in a half-cylinder with o~ domain space as base), and the kernels of ~1/2 sin (tK l/2 ) and K-1/2 cos(tK 12) • These last two are elementary solutions of the wave equation - d~ u = Ku, and are very :important in quantum field theory: The first is the well-known corrmutator function used to solve the cauchy problem, and the second is the synrnetrized vacuum two-point function, G(l) (t,x o ,y) , which is central to the calculation of physical quantities such as energy density for a quantum field subject to external potentials. In those calculations, those terms of various derivatives of G(l)
184
STEPHEN FULLING
which diverge as y -+ Xo must be subtracted off in a well-defined way to leave a finite and calculable renormalized remainder [8,6,4,29,30,13J. This would be facilitated by knowing precisely how the singular terms arise out of an eigenfunction expansion -- hence my interest in approaching the two-point function from the direction of the local behavior of the spectral projections. I shall now show how this effective, or Ill2an-asymptotic, expansion of dpoo (for instance) can be obtained directly from a study of the eigenfunctions of K at large :\. Since we know (from theorems about the heat kernel, for example) tJ:1at the series gepends on V only locally, it suffices to consider any potenti~l V, on a manifold M, which coincides \'lith V on ~ neighborhood of Xo where M and M can be locally identified. I choose M = *- and V £ ex', and henceforth drop the tildes. Now we have a routine quanttm1 scattering prob:iem. Introduce eigenfunctions ¢w normalized by their behavior at infinity, so that 1 [¢ (x)@¢(y)+¢(x)0¢(YJ]dw --~ -2 T[ w ID -(e -w 'k
(A =: w 2 > 0).
WAj (x) dpJ (A;X O) wAk(y)
L
ciliA (x,y)
j,k
Restrict attention for a moIll2nt to the scalar case. Vw = (jJ2 Wcan be approximated this way:
A basis of solutions of
- W" +
w(x)
=
Vo
[L
2 1/2 p- v Y2)x)]exp[ip
V=O
tXo
+ O(w- 2Vo - l ) ,
Vo
L
p
-2v
v=o
Y )x') dx'] 2
p = ± w •
The Y's are found by solving a recursion relation, and the first few are Yo
=
=-
Y2
1 ,
!
Y4
V ,
=
%(V"
- V
2
)
•
They are tabulated [5] up to Y20 - - which has 137 terms. This form of the WKB approximation has been developed especially by Froman [11]. Its crucial feature is that the amplitude of the approxination is purely local; integrals over x appear only in the phase. (This is rather surprising: It says that the relation between global orthonormalization of eigenfunctions and the values of the functions at X o is asymptotically determined by the potential and its derivatives at Xo alone.) From these equations it easily follows that dp
00
(:\,x o)
'V
1 TI
L P v
00
v
-2v (x o) w dw,
a series dbtained by formally taking the reciprocal of the series LY2vW-2V ; in other words, the spectral density dp 0/dw for this problem is asymptotically equal to the square of the WKB amplitude function. The other densities are related similarly to the derivative of the WKB expression.
°
I have worked out the analogous local WKB expansion for the vector case. basic ansatz is that the eigenfunctions in the basis satisfy W' (x) = ip Np (x) W(x)
,
where
N
P
'V
-s
LpNs
s=O
Then one shows that W(X)
'V
A(x) v(x)
Ilvil = 1
,
where A or any power of A has a local expansion coming from A(x)-2
'V
L w- 2V Y 2v v
=~
,
The
LOCAL ASYMPTOTICS OF CONTINUUM EIGENFUNCTION EXPANSIONS
185
a diagonal matrix element of the even part of N. Again matrix elements of dpoo/dw can be identified with A(X o )+2, etc. I have also written computer programs which calculate the N' s and the coefficients in the expansion of dW 00 up to about s = 14 (v = 7) , where the m:nroer of tenns becorres too large to print out feasibly. (N 1 contains 127 terms of dirrension [length]-14, ranq.ing fr-om V(12) through (V,)2 V(3) V', etc., to V7 .) In particular, I can report that the coefficients left undetermined by Gilkey [16] in his calculation of EO 0 are a = 21, b = 28. 4 Note that in this scattering problem on M = f the series are truly asymptotic. For an operator locally equivalent to the scattering operator, but on one of the other manifolds, or with an unbounded potential, the series will have the weaker significance I described earlier. That is, the remainders are not small compared to the terms in the series, and they depend on nonlocal information such as how far Xo is from the boundary. However, this nonlocal contribution is oscillatory, and that is why it doesn't contribute to the singularities of the Green functions, which arise in limits where the spectral densities are integrated against a very slowly varying function. The nonlocal effects would show up in a WKB treatment through reflected waves, turning points, and quantization conditions - all the complications for WKB of a potential which is not a S!IDOth function of compact support on the whole real line. I have extended the WKB calculation of the spectral rreasures to the case of scalar functions on the half-line with the most general boundary condition, ~'(O) = K ~(O). The eigenfunction now has a reflected wave equal in strength to the incident wave. I find
iWN
+ Re [
+K
'wN~
1
+w -
K
exp(2i
J~o ~
v=l
w-
2v
+l Y2)x') dx') e
2iwx ] o } .
Note that as x 0 (distance from the boundary) becomes large, the boundary correction term does not become small, but it does oscillate faster and faster. I 2iwx expand everything except e o in inverse powers of w and calculate the Laplace transform term by term to get the boundary correction to the heat kernel; this can be done in closed form (for tirre derivatives of H) in tenns of Hermite functions, which do falloff as exp (- x 0 2 It) away from the boundary. Integrating that result over Xo ' I can recover the known contribution of an endpoint to the integrated "trace" of the heat kernel [16]. It is clear that a similar analysis would apply near a point where some derivative of V has a jLU11p discontinuity, giving rise to a reflected wave in the WKB approximation. This raises the prospect of a unified theory of boundaries and coefficient singularities, with a boundary appearing as an extreme case of a singularity. I do not claim that this kind of calculation is the most efficient way of determining the local invariant quantities; the traditional rrethods [10,1,6,15,16,14, 31] based on integral kernels may well be better. Nor does it replace the theorems which establish the universal nature of the singularities in the first place. I do assert that my treatment sheds light on the origin of these quantities, and also paves the way for the renormalization program in quantum field theory which I described. (Also, the WKB approximations for eigenfunctions are useful in their own right and should hardly be considered part of the expense of this particular application.)
186
STEPHEN FULLING
I conjecture that the approach can be extended to higher-dimensional manifolds by combining the higher-dirrensional WKB approxirrBtion of Maslov [25] with the FrOman idea that the amplitude can and must be kept local to arbitrarily high order. I am grateful to J. stuart Ilc::M'ker and Michael Taylor for introducing me to important literature. This research has been supported by National Science Foundation Grants Nos. PHY79-15229 and PHY77-27084. REFERENCES
[1] Balian, R. and Bloch, C., Distribution of eigenfrequencies for the wave equation in a finite domain. I, Ann. Phys. (N.Y.) 60 (1970), 401-447. [2] Baltes, H.P. and Hilf, E.R., Spectra of Finite Systems, Bibliographisches Institut, Mannheim, 1976. [3] Br=ell, F.H., Extended asymptotic eigenvalue distributions for bounded domains in n-space, J. Math. Mech. ~(1957), 119-166. [4] Bunch, T.S., Christensen, S.M., and Fulling, S.A., Massive quantum field theory in two-dimensional Robertson-Walker space-time, Phys. Rev. D 18 (1978), 4435-4459. [5] Campbell, J.A., Computation of a class of functions useful in the phaseintegral approxirrBtion. I, J. Comput. Phys. 10 (1972), 308-315. [6] Christensen, S.M., Vacuum expectation value of the stress tensor in an . arbitrary curved background: The covariant point-separation method, Phys. Rev. D 14 (1976), 2490-2501. [7] Clark, C., The asymptotic distribution of eigenvalues and eigenfunctions for elliptic boundary value problems, SIAM Rev. ~ (1967),627-646. [8] DeWitt, B.S., Quantum field theory in curved spacetime, Phys. Reports 19 (1975), 295-357. [9] Duistermaat, J.J. and Guillemin, V.W., The spectrum of positive elliptic operators and periodic bicharacteristics, Invent. Math. 29 (1975), 39-79. [10] Friedlander, F.G., The Wave Equation on a Curved Space-Time, Cambridge Univ. Press, Cambridge, 1975. [11] FrOman, N., Outline of a general theory for higher order approxirrations of the JWKB-type, Arkiv Fysik E (1966), 541-548. [12] Fulling, S.A.,and Narccwich, F.J., A basis for the local solutions of an elliptic equation, J. Math. Anal. Appl., to appear. [13] Fulling, S.A., Narccwich, F.J., and Wald, R.M., Singularity structure of the two-point function in quantum field theory in curved spacetime. II, to appear. [14] Gel' fand, LM., and Dikii, L.A., Asymptotic behavior of the resolvent of SturmrLiouville equations and the algebra of the Korteweg-deVries equations, Usp. Mat. Nauk 30:5 (1975), 67-100 [Russ. Math. Surv. 30:5, 77-113]. [15] Gilkey, P.B., The spectral geometry of a Riemannian manifold, J. Diff. Geom. 10 (1975), 601-618. [16] Gilkey, P.B., Recursion relations and the asymptotic behavior of the eigenvalues of the Laplacian, Compos. Math. ~ (1979), 201-240. [17] Greiner, P., An asymptotic expansion for the heat equation, Arch. Rat. Mech. Anal. 41 (1971), 163-218. [18] Hadamard, J.S., Lectures on Cauchy's Problem in Linear Partial Differential Equations, Dover, New York, 1952.
LOCAL ASYMPTOnCS OF C()NTINUUM EIGENFUNCTION EXPilNSIONS
187
[19] Hormander, L., On the Ries z rrt2ans of spectral functions and eigenfunction expansions for elliptic differential operators, in Belfer Graduate School of Science Annual Science Conference Proceedings: Some Recent Advances in the Basic Sciences, Vol. 2 (1965-66), ed. by A. Gelbart, Yeshiva Univ., New York, 1969, pp. 155-202. [20] Hormander, L., The spectral function of an elliptic operator, Acta Math. 121 (1968), 193-218. [21] Kodaira, K., The eigenvalue problem for ordinary differential equations of the second order and Heisenberg's theory of S-matrices, Am. J. Math. 71 (1949), 921-945. [22] Kodaira, K., On ordinary differential equations of any even order and the corresponding eigenfunction expansions, Am. J. Math. 72 (1950), 502-544. [23] Kulkarni, R.S., Index Theorems of Atiyah-Bott-Patodi and CUrvature Invariants, Presses univ. Montreal, Montreal, 1975. [24] Levitan, B.M., On the asymptotic behavior of the spectral function of a selfadjoint differential equation of the second order and on expansion in eigenfunctions. II, Izv. Akad. Nauk SSSR, Ser. Mat., 19 (1955), 33-58 [Am. Math. Soc. Transl. (2) 110 (1977), 165-188]; and-related papers. [25] Maslov, V.P. and Fedoryuk, M.B., The Quasiclassical Approximation for the Equations of Quantum Mechanics, Nauka, Moscow, 1976 [Russian]. [26J McKean, H.P. and Singer, I.M., Curvature and the eigenvalues of the Laplacian, J. Diff. Geom. ! (1967), 3-69. [27] Naimark, M.A., Linear Differential Operators, English ed., F. Ungar, New York, 1968. [28] Titchmarsh, E.C., Eigenfunction Expansions Associated with Second-order Differential Equations, Part One, 2nd ed., Oxford Univ. Press, Oxford, 1962. [29] Wald, R.M., The back reaction effect in particle creation in curved spacetirrt2, Commun. Math. Phys. 54 (1977), 1-19. [30] Wald, R.M., On the Euclidean approach to quantum field theory in curved spacetirrt2, Commun. Math. Phys. 70 (1979), 221-242. [31J Widom, H., A complete symbolic calculus for pseudodifferential operators, Bull. Sci. Math. 104 (1980), 19-63.
This page intentionally left blank
This Page Intentionally Left Blank
Spectral Theory of Differential Operators I. W. Knowles and R. T. Lewis leds.) © North-Holland Publishing Company, 1981
SOME OPEN PROBLEMS ON ASYMPTOTICS OF m-COEFFICIENTS Charles T. Fulton Dept. of Mathematical Sciences Florida Institute of Technology Melbourne, Florida 32901
Some open questions concerning asymptotic expansions of Weyl-Titchmarsh m-coefficients for second order equations on rays and lines in the complex A-plane are described for problems which involve the eigenparameter linearly in one boundary condition. For the classical Sturm-Liouville problem on [a,oo)
(1)
-(py' )'+ qy = AY, y'(a)
=
(2)
0,
with p>O, qEL 1 ,loc[a,oo) and P'LACloc[a,oo), the Weyl-Titchmarsh m-coefficient in the limit point case is the function meA) which is uniquely defined for Im\ i 0 by the requirement 8 (x) + meA) ¢\(x) f:L [a,oo), (3) 2 where {¢\' ¢\ \ are the solutions of equation (1) defined by the initial conditions
¢\(a) ( p(a)¢\ (a)
8\(a)
\
p(a)8\(a>j
=
(-1 0) o
+1
(4)
The investigation of the asymptotic behaviour of meA) for complex \ was initiated by Einar Hille who proved ([9; §10.2, Theorem 10.2.1]) in the case p=l, and for q continuous, that (5 )
m( \)
uniformly for GEG o := {elo~ 8 ~ Tf-O, -rr+o~ e ~-o}, 0>0_ (6) W.N. Everitt «(4; p. 447, Equa. (4.5)]) improved on Hille's result, obtaining under the above assumptions on p, q that meA) = i/(p(a)\)1 / 2 + 0(1/1\1), (7) uniformly for GEG o' (Here the branch of \ is understood to be taken on the positive real \-axis.) Improvements and refinements of Everitt's results in [4]have been made by Everitt and Halvorsen [51and recently by F.V. Atkinson [l]. The technique of estimation employed by Everitt relies on the use of asymptotic formulae for the solutions ¢\ and 8\ as given, for example, by Titchmarsh~; §1.71 For the case of the similar problem with a A-dependent boundary condition at the left endpoint, 189
CHARLES T. FULTON
190
-y" + qy = :\y,
xE[a,oo),
(8 ) ( 9)
(a 1 y(a) - a 2 Y'(a» = :\(aiy(a) - a Y'(a», a:= aia2 - a a 1 > 0,
2
(10)
2
with qEL I 1 ~,oo), the basic expansion theory has been given , oc by the author in [6], and the Weyl-Titchmarsh m-coefficient in the limit point case is again uniquely characterized by the requirement, (11 )
where
{
S:\} are now defined by the initial conditions
(
(~~/a\
~l-al:\
(12)
aI/a)
By relying on first order formulae for the asymptotic behaviour of
Let q belong to the L.P. case at 00. 3 2 m(:\) = (1/0.:\) + 0(1/1:\1 / ), as IAI-+oo,
Then (13)
18
(ii)
:\ = IAle , uniformly for 8EG ' 3/2 o m(lHiv) = (-i/av) + 0(1/lvI ), as
uniformly for )J E [-K,K J, O
VE[-K,-6]U[6,KJ,
OPEN 1)
p
Iv 1-+ 00 ,
(14)
-++00,
(15)
O<6
PROBLEMS
Under further assumptions on q, to find the constant c 1 ' terms of q,such that
in (16)
as :\-+00 2)
on rays, uniformly for [)
E
8 , 6
By employing higher order formulae for
=
c1 3/2 +
_1_ + a:\
A
c __ 3_ + ... :\5/2
(17)
under successively stronger assumptions on q. In particular, is there an iterative scheme for generating the formulae for c directly in terms of q? The same question applies, of course~ to possible extensions of the classical case (7) considered by Everitt. Here, Atkinson [lJ uses a Ricatti method to get c l · 3)
Can similar results on vertical lines and horizontal lines, extending parts (ii) and (iii) of Theorem 1, be obtained?
191
SOME OPEN PROBLHMS ON ASYMPTOTICS OF M-COHFFIClcNTS
4)
If the left endpoint is taken to be limit circle, then by making use of 'end conditions' at the L.C. endpoint the author has shown in [6 ]that the solutions ¢A and 8 A can be defined at the singular L.C. endpoint so that the associated m-coefficient for the problem on (a,co), -ooca
1m m(iv)
I =
(18)
1/11.
How does one obtain sharper results like those in Theorem 1, and higher order results similar to those just suggested when the left end is regular, by making suitable assumptions on q near the L.C. endpoint? Here, some recent results of Atkinson and Fulton [2], [3] on the asymptotics of eigenvalues and solutions may provide necessary preliminary information to take the place of more standard asymptotic results when the left endpoint is regular. ~)
Can the classical m(A)-function of (3) above be modified by putting ~A(x)
=
6e ll )¢A(x) and Bilex)
(19)
for some suitable 0(11) such that
(20)
mOl satisfies lim v
11m m(iv) I
=
const < 007
v +00
(21)
If so, what change in (7) will such a re-normalization effect for ~(iI), and could such a re-normalization (together with a corresponding re-normalization of the spectral function) be used to advantage in simplifying classical proofs of convergence results for the associated eigenfunction expansion? For further detail on this pOSSIbility we refer to the author'S paper [6; pp. 28-30] where relatec questions are raised. References (1)
Atkinson, F. V., On the location of the Weyl circles, Proc. Roy. Soc. ~din., Sec. A, to appear.
(2)
Atkinson, F.V. and Fulton, C., Asymptotic Formulae for eigenvalues of limit circle problems on a half line, Annali di Math. Pur. ed. Applicata, to appear.
(3)
Atkinson, F. V., and Fulton, C., Asymptotic Formulae for elgenvalues of limit circle problems on finite intervals, SUbmitted.
(4)
Everltt, W. N., On a property of the m-coefficlent of a second order linear differential equation, J. London Math. Soc. 4 (197~)
443-457.
192
CHARLES T. FULTON
(5)
Everitt, W. N. and Halvorsen, S. G., On the asymptotic form of the Titchmarsh-Weyl m-coefficient, Applicable Anal. 8 (1978) l53-169.
(6)
Fulton, C., Singular eigenvalue problems with eigenvalue parameter contained in the boundary conditlons, Proc. Roy. Soc. ~din. 87A (l980) l-34.
(7)
Fulton, C., An integral equation iterative scheme for asymptotic expansions of spectral quantities of regular SturmLiouville problems, Jour. Integral Equas., to appear.
(8)
Fulton, C., Asymptotics of the m-coefficient for eigenvalue problems with eigenparameter in the boundary conditions, Bull. London Math. Soc., to appear.
(9)
Hille, E., Lectures on ordinary differential equations (Addison-Wesley, Reading, Mass., 1969).
(10)
Titchmarsh, E. C., Eigenfunction expansions associated with second-order differential equations: Part I (Oxford University Press; 2nd edition, 1962).
Spectral Theory of Differential Operators I.W. Knowles and R. T. Lewis leds.) © North·Holland Publishing Company, 1981
SINGULAR LINEAR ORDINARY DIFFERENTIAL EQUATIONS WITH NON-ZERO SECOND AUXILIARY POLYNOMIAL Richard C. Gilbert Mathematics Department California State University Fullerton, California U.S.A. A homogeneous linear ordinary differential equation is studied under the assumption that the coefficients are holomorphic and have asymptotic expansions in a sector of the complex plane. Asymptotic formulas are determined for that part of a basis which corresponds to a root of certain auxiliary polynomials. When the results are applied to a formally symmetric operator, a new situation turns up in which the number of integrable-square solutions for A in the upper half-plane can differ by two or more from the number of integrable-square solutions for A in the lower half-plane. INTRODUCTION Let S be an open sector of the complex plane which has vertex at the origin and contains the positive real axis. Consider the equation Ly ~ L n Cl (x)y(r) = 0 (1) r=O n-r ' where ClO(X) = 1, the Cl n-r (x) are holomorphic for XES, 0 < Xo ~Ix I < and for r = 0, 1, ••• , n , x- k Cl n-r (x) ~ =;-0 (2) - Cl n-r, k as x -> in each closed subsector of S. For h = 0, 1, ... , let n r Ih(j..I) = Lr=O Cl n-r, h ).J ( 3) 00
,
00
We shall call the
Ih(j..I)
the auxiliary
polynomials.
Solutions of equations of type (1) with various assumptions on the auxiliary polynomials have been studied by Orlov [6], Kogan and Rofe-Beketov [5], and Gilbert [1,2,3J. In the present article it is assumed that IO(j..I) = 0 has a root S of multiplicity m ~ 2, I 2(S) f 0, and Iir)(S) = 0 for r = 0, 1, ••• , K - 2, where K is an integer (depending on S) for which m/2 + 1 < K < m and I (K-l)(S) f 0 if K < m. 1 Solutions of (1) corresponding to S are determined. The results are applied to the study of the square integrable solutions of 9,y = AY, where A is a complex number, and 9,y is a formally symmetric linear ordinary differential operator. A new situation turns up in which the number of square integrable solutions for IA > 0 can differ from the number for IA < 0 by an arbitrary pre-assigned integer. 193
194
RICHARD C. GILBERT
FUNDAMENTAL MATRICES Theorem 1. Suppose that A(x) is an m by m matrix (m ~ 2) which is holomorphic for x s S, 0 < xo ~ Ixl < 00, where S has positive central angle less than TI/(q + 1) for a certain non-negative integer q. Suppose that A() x ~ Lr=O Ar x-r as x + in each closed subsector of S. Suppose that AO ha~ distinct eigenvalues d , d , ••• , d 2 1 m and that Q is a matrix for which Q- AQ = diag(d 1 , d2 , ••• , dm). Let T be any closed subsector of S. Then for x s T, Xo ~ Xl ~ Ixl < Y' = xq A(x) Y has a fundamental matrix of the form Y(x) = QM(x) exp [R(x) + H log xJ ,where M(x) is holomorphic for x s T , oo Xl -< Ixl < 00, M(x) ~ Lr-_ O Mr x- r as X + in T, and MO = E. (We use any convenient branch of log x.) H is a diagonal matrix whose diagonal elements are certain complex constants n1 , n2' ••• , nm • R(x) is a diagonal matrix whose diagonal elements rs(x), s= 1, 2, •.• , m, are polynomials of the form , q+1 ss q q-l Jk rs(x) = [ds/(q + I)J x + ~~1 /qJx + aq_ x + ••• + a x, where the all 1 1 are the elements of All = Q Al Q, and aI' a2 , ••• , aq_1 are certain complex constants which depend on s. If the elements of AO are all zero except for those on the upper off-diagonal, each of which is 1, and the elements of the last row, which might be non-zero, then we can take Q to be the matrix whose k-th column for k = 1, 2, ••• , m consists of the transpose of 2 m-l (l,dk,dk,···,d k ). The proof of Theorem 1 is similar to that of [3, Theorem 3J. 00
00
00
,
Theorem 2. Suppose that A(x) is an m by m matrix (m ~ 2) which is holomorphic for x € S, 0 < Xo ~ lxl < oo, where S has positive central angle less -r than TI. Suppose that A(x) ~ Lr=O Ar x as x + in each closed subsector of S, where AO is a matrix all of whose elements are zero except those on the upper off-diagonal, each of which is 1, and those on the diagonal, each of which is 6 (which is possible not zero). Suppose that for r > 1, 00
00
'k
'k
Ar = [a r J J;,k=l and that ai = 0 for 1 ~ j ~ m, 1 ~ k < K - 1 , where K is a fixed interger such that (m/2) + 1 < K < m. If K < m, suppose also that mk ml - , a 1 o. Suppose that a2 1 o. Let DO be an m by m matrlx whose elements l above the last row are all zero except for the elements on the upper Off-diagonal, each of which is 1. If m = 2, let the last row of DO consist of (a~1 , 1 + a~2). In the case m = 2M, M > 2, K = M+ 1, let the last r
00
•
,
00
SINGULAR LINEAR
ORDlN.~RY
195
DlI,'j·'ljRliNn.·1L I:QUATIONS
P is a positive integer, and all the elements of MO are zero except those in the first row, each of which ;s 1. If m = 2, U(x):= 0, F = diag(d ,d ),p = 1. 1 2 If m > 2, F is a diagonal matrix whose diagonal elements are certain complex constants, while U(x) is a diagonal matrix whose diagonal elements us(x), s = 1, 2, ••• , m, are functions of the form us(x) = d p s-l xE/ p + b _ x(E-1)/p + ••• + b x1/p, where E = P - d, d is s E 1 1 a positive integer less than p, and b1 , b , ••• , bs _ 1 are certain ~omplex 2 numbers which depend on s. Any convenient branches of log x and x /p may be used. For m = 2, Theorem 2 is proved like [1, Theorem 10J. For m > 2, Theorem 2 is proved like [3, Theorem 4J with use of [4, Theorem 4J and Theorem 1. A BASIS FOR THE SOLUTIONS OF
Ly = 0
Theorem 3. Suppose that S has positive central angle less than ]f. Consider equation (1) with coefficients a (x) satisfying the hypotheses after n-r equation (1) and with the Ih(v) given by (3). Suppose IO(w) = 0 has a root S of multiplicity m ~ 2. Suppose rir)(S) = 0 for 0 ~ r ~ K - 2 , where K is a fixed integer such that m/2 + 1 < K < m. If K < m, suppose that Ii K- 1 )(6) f O. Let I (6) t- O. Let a =--[11;] ri K- 1 )(S)][(K-1)! I6 m)((3l]-1 , 2 b = -em! I (S)]' [I6m)((~)]-I. Let DO be an m by m matrix whose elements 2 above the last row are all zero except for the elements on the upper off-diagonal, each of which is 1. If m = 2, let the last row of DO consist of (b, 1 + a). In the case m = 2M, M~ 2, K = M+ 1, let the last row of DO consist of (b, 0, .•. , 0, a, 0, ••. , 0), where a is in the K-th spot. In all other cases of m and K, let the last row of DO consi st of (b, 0, ••• , 0). Suppose that the eigenvalues d , d , ••• , d of DO are distinct, and in the m 1 2 case m = 2, suppose that they do not differ by an integer. Let T be a closed subsector of S. Then, if T has sufficiently small positive central angle, corresponding to S there are functions fI(x), f (x), ••• , fm(x) which are 2 part of a basis for the solutions of (1) and have the form (4) f.(x) = [1 + h.(x)J exp[8x + u.(x) + Tl. log xJ, J J J J j = 1, 2, •.• , m. Here, hj(x) is holomorphic for x s T, Xo ~ Xl ::.. Ix I < and hj(x) = 0(1) as x -+ in T. If m = 2, Uj(x) := 0 and Tlj = dj for j = 1, 2. If m > 2, u.(x) has the form (t:-1)/p +... + b x l/p , (5) u.(x) = d. Jps -1 xsIp + b x' J J [-1 1 where p is a positive integer, t: = p - d d is a positive integer less than p, and b , b , ••• , b _ depend on j. Any convenient branches of s 1 1 2 x1/ p and log x may be used. 00
00
Proof. By [3, Section 2, Theorem 1 and Remark 1J and a modification of [1, Theorem 3J, we can show that corresponding to S there is a subsystem to
196
which Theorem 2 applies.
RICHARD C. GILBERT
Theorem 3 now follows from [3, Remark 2J.
Theorem 4. Suppose S has positive central angle less than TI. Consider (x) satisfying the hypotheses after (1) and equation (1) with coefficients a n-r () n n-l(.)n-r r with the Ih(~) given by (3). Suppose 10 w = w + ~ r--0 -1 POw n-r, where the P 0 are real. Let HO(v) = in IO(-iv). Suppose 6 = s-it is n-r, ( ) a root of multiplicity m > 2 of IO(w) = O. Suppose I l r (6) = 0 for o < r < K - 2 , where K is a fixed integer such that (m/2) + 1 < K < m. If K < m, suppose that Ii K- 1) (6) f O. ~et 1 (w) = J (11) - i- n A~ where 2 2 lfl J (w) is a polynomial in ~, and A = pe is a complex number such that 2 1 (6) f O. Let the f .(x), j = 1, 2, ••• , m, be given by (4) with x real 2 and those branches of J x1/p and log x chosen which are real for real x. Then, the following are true: (A) If s < 0, f1' ••• , fm are all in L2; if s > 0, f , .•• , fm are all not l in L2 • (B) Suppose s = 0 and m is odd. If H6 m) (t) > 0, then for each fi xed 8. 0 < 8 < TI, for all p sufficiently large exactly (m - 1)/2 of the solutions (4) are in L2; and for each fixed 8, -TI < 8 < 0, for all p sufficiently large exactly (m + 1)/2 of the solutions (4) are in L2. If H(m)(t) < O. the above situation is reversed. o (C) Suppose s = 0 and m is even. For each fixed 8, 0 < 8 < TI or -TI < 8 < 0, for all p sufficiently large exactly m/2 of the solutions (4) are in L2. The proof uses Theorem 3 and is similar to that of [3. Theorem 8J. SQUARE INTEGRABLE SOLUTIONS OF Ly = 0 Theorem 5. Suppose that S has positive central angle less than TI. Consider (x) satisfying the hypotheses after (1) equation (1) with coefficients a n-r r and with the Ih(w) given by (3). Suppose 10 (11) = wn + ~n=ol(_i)n-r P 0 I1 • n rn-r. where the Pn-r,o are real. Let HO(v) = i IO(-iv). Suppose . I (w) = J 2(1l) - i- n A, where J 2 (11) is a polynomial in w, and A = pe 18 2 is a complex number. For each root y of Ho(v) = 0 of multiplicity m ~ 2. suppose that lir)(-iy) = 0 for 0 ~ r ~ K - 2. where K is an integer (depending on y) for which (m/2) + 1 < K < m; and if K < m. suppose that (K-1) . ( _ II (-lY) f O. Let t 1 , t 2, •.•• ta be the real roots of HO v) - 0 such that for 1 ~ s ~ a, ts has odd multiplicity ms ~ 3, and H~o)(ts) < 0 , where a = ms' Let Tl , T2 •••• , Tb be the real roots of HO(v) = 0 such that for 1 < s < b, Ts has odd multiplicity Ms ~ 3, and H~o)(Ts) > 0 • where a = ms' Let v be the number of real roots of HO(v) = 0 which have
SINGULAR LINEAR ORDINAR Y DIFFERENTIAL EQUA nONS
197
multiplicity 1. Suppose N is the number (counting multiplicities) of real rootsof HO(v) = 0 which have even multiplicity together with the number (counting multiplicities) of non-real roots of HO(v) = O. Then, equation (1) has a basis formed by the functions (4) and by the functions (38) of [3J, and for this basis the following are true: (A) For each fixed e, 0 < 8 < IT , for all p sufficiently large exactly b Z-I[Za - s=1 ms + Ls =1 Ms + b - a + N] + v of the functions in the basis are in LZ. (B) For each fixed e, -IT < e < 0, for all p sufficiently large exactly Z-I[Z:=1 ms + Z~=1 Ms + b - a + NJ + v of the functions in the basis are in LZ. Z The proof uses Theorem 4 and [3, Theorem 6] to count the L solutions. Theorem 6. Let S have positive central angle less than IT. Let a be a positive integer. Let b be an integer, 0 < b < a-I. Let d be a nonnegative integer. Let r 1 , r Z' ••• , r a , r a +1 , ••• , r a+b , r a+b+1 , " ' 1 r a +b+d be real numbers with the following properties: r 1 < r Z < ••• < ra ; if b > 0, then rj < r a +j < r j +1 for 1 ~ j ~ b; if d > 0, then ra < r a+b+ 1 < ••• < r a+b+d • Let t3u-Z t 3u _1 = t3u = ru for 1 < u ~ a , and if b > 0 or d > 0 , let t 3a +u = ra+u for 1 ~ u ~ b + d. Let n = 3a + b + d. Suppose the functions P (x), m = 0, 1, ••• , n, are holomorphic for XES, 0 < Xo ~ Ixl < 00, m Z and real-valued for real x. Suppose that Po(x) =: x , and that for Zoo -k m = 0, 1, ••• , n, Pm ( x) ~ x Zk=O Pmk x as x + in each closed subsector of 5, where Pm1 = 0 for 1 ~ m ~ n, and PmO is given by the following equations: P10 = zt-, 1 -< j -< n; PZO = Lt.J tk ' 1 -< j < k -< n; J P30 = Ltj tk tm ' 1 ~ j < k < m ~ n; ••• ; PnO = tl tz ••• tn' Suppose £y = L~=oik Qk y, where £Zr y = (Pn_zr(x)y(r)) (r), £Zr+l y = (I/Z) {(P n- Zr - 1(x) y(r))(r+l) + (P n- Zr - 1(x) y(r+l))(r) }. Then, the following are true: (il £y is a formally symmetric ordinary differential operator on [x O' co). (ii) The equation £y = AY, where A = pe i8 , can be written in the form of equation (1) with coefficients a (x) satisfying the hypotheses after (1). I ' h I ( ) n n_~-r( t 1S true t at 0 lJ - lJ + Zr=o -1.)n-r Pn-r,O lJ r -_ ( lJ-S I )3 •.• ( lJ-S a )3( lJ-S a+l )••• (lJ-B a+b+d ), where Ss = irs llr)(Ss) = 0 for r = 0, 1, and 1 ~ s ~ a ; Iz(lJ) = J 2(lJ) - i- n A, where J 2(lJ) is a polynomial in lJ. Hence, Theorem 5 applies. (iii) £y = AY has a basis formed by the functions (4) and by the functions (38) of [3J, and for this basis the following are true: (A) For each fixed 8, 0 < G < IT , for all p sufficiently large exactly b + 1 + [(a - b - 1)/ZJ + 2[(a - b)/Z] + b + d of the elements in the basis are in LZ. (Here, [J stands for the largest integer function.) (B) For each fixed e, -IT < 8 < 0, for all p sufficiently large exactly 00
198
RICHARD C. GILBERT
2(b + 1) + 2[(a - b are 1. n L2 •
1)/2J + [(a - b)/2J + b + d of the elements in the basis
Conclusions (i) and (ii) follow as in the proof of [2, Theorem BJ. from Theorem 5.
(iii) follows
Remark. Theorem 6 is a simplification of Theorem B of [2J in that Po(x) = x2 rather than Po(x) = x3 , and it is not necessary to specify the P by m2 equations (52), (53) of [2J. Note, however, that when using Theorem 6, seven is still the minimum order of the operator for which the number of L2 solutions for IA > 0 differs by two from the number of L2 solutions for IA < O. REFERENCES [lJ [2J [3J [4J [5J [6J [7J
Gilbert, R.C., Asymptotic formulas for solutions of a singular linear ordinary differential equation, Proc. Roy. Soc. Edinburgh. Sect. A 81 (1978), 57-70. Gilbert, R.C., A class of symmetric ordinary differential operators whose deficiency numbers differ by an integer, Proc. Roy. Soc. Edinburgh. Sect. A 82 (1978), 117-134. Gilbert, R.C., Integrable-square solutions of a singular ordinary differential equation. to be published by Proc. Roy. Soc. Edinburgh. Sect. A. Gilbert, R.C., Shearing transformation ofa linear system at an irregular singular point, to be published by Math. Proc. Cambridge Philos. Soc. Kogan, V.I. and Rofe-Beketov, F.S., On the question of the deficiency indices of differential operators with complex coefficients, Proc. Roy. Soc. Edinburgh. Sect. A 72 (1975), 281-298. Orlov, S.A., On the deficiency index of linear differential operators, Ookl. Akad. Nauk SSSR 92 (1953), 483-486. Warsow. W., Asymptotic expansions for ordinary differential equations (Interscience, New York, 1965).
Spectral Theorv of Differential Operators I. W. Knowles and R. T. Lewis leds.) © North·Holland Publishing Company, 1981
HIGHER DIMENSIONAL SPECTRAL FACTORIZATION WITH APPLICATIONS TO DIGITAL FILTERING R. Kent Goodrich Karl E. Gustafson University of Colorado Boulder, Colorado, USA
A key tool in the theory of digital filtering in one dimension is a certain general spectral factorization. The lack of such factorization has been a major impediment in the development of a digital filtering theory in higher dimensions. We give here a general method for such factorization in any number of dimensions.
INTRODUCTION AND BACKGROUND Two and three dimensional filters are currently under much investigation in the electrical engineering community and are central to many array processing applications. In the one dimensional theory one employs a general factorization of a certain spectral density associated with the process under consideration into the product of an inner function and an outer function. Among all filters which produce the same gain at each frequency, the outer function corresponds to the filter producing that gain with the minimum group and phase delays. Outer functions have no zeros in the upper half plane. Thus all such zeros in the Hardy function being factored have been absorbed into the inner function. The latter is essentially and in many cases a Blaschke product. Because the zeros of functions of more than one complex variable are generally continua, there has been difficulty in extending the filtering theory to more than one dimension. Our method is neW and apparently the first general inner-outer spectral factorization in higher dimensions. Its abstractness, coming from a functional analytic approach and from considerations of stochastic processes in quantum mechanics, has not as yet been tested as to direct applicability to filtering problems. For the moment, it may be viewed as the beginning of a new theory of inner and outer functions in higher dimensions. It also has important implications in higher dimensional approximation theory. We hope to show in this paper its possible implications to digital filtering in higher dimensions. Further details of the analysis and full proofs of a number of the results given here may be found in a paper to appear [1]. A preliminary announcement of some of these results was given in [2], where the emphasis was on the relationship to higher dimensional purely nondeterministic stochastic processes. It should be stressed here that the higher dimensionality is in the parameter variable, and not in the vector valued random variable, which has been and usually can be generalized from the one dimensional scalar range to finite, infinite, and matrix valued ranges. Some further details, especially as to the relations to regular representations of arbitrary groups and to support questions for generating cyclic vectors, may be found in [3]. There also the connection to fundamental approximation problems is emphasized. In this present paper we wish to describe somewhat briefly the results given in [1], also [2] and [3], and moreover to attempt to place them in the context of filtering theory, where their eventual implementation may be of significant practical, beyond conceptual, value. Their connection to the spectral theory of differential operators, the subject of this conference, is threefold. First, as
199
200
R. KENT GOODRICH and KARL GUSTAFSON
is well known, modeling physical systems subject to random inputs yields solutions of ordinary differential equations in terms of realizable convolution filters in a wide variety of situations, such as in the theory and application of Kalman filters. Second, as was established in [4], all square integrable white noise processes are unitarily equivalent to quantum mechanical momentum evolutions. In particular, such evolutions are generated by first order partial differential operators with absolutely continuous spectra the whole real line and with an additional spectral requirement on the spectral density that corresonds to the optimal gain property mentioned above (this will be explained below). Thirdly, these questions can be posed in terms of boundary value problems for the Laplacian on the upper half plane, or, in the higher dimensional cases, the Laplacian in half spaces, quadrants, and other configurations. DIGITAL FILTERING AND FACTORIZATION Two excellent references for these topics are [5] for the filtering theory and [6) for the function theory. Much of what we say here may be found therein, although we will take a slightly different point of view here, stressing the most elementary connections between the filtering theory and the function theory, from the point of view of spectral theory, in order to make the connections to our approach in [1].
Given an input process
X
or
, parametrized here by a one dimensional parameter
t
,which may be time or space and which may be discrete or continuous, a
L : X + Y t t
linear filter is a linear operator X
L2
or both, whenever need be, without further specification thereof.
L1
t
We will imagine all processes as in
t
The resulting transformed process
Y
t
on the space spanned by the
is called the output process.
filter is required to have the "time invariance property"
The
L
Because of this latter property one immediately sees a connection to group representation, which is one facet of our approach in [1). Moreover, for higher dimensional filtering applications the parameter t should be repla ced by a g for a general group, and in particular by v for a two or three dimensional space variable in the group Often
X
t
Y
t
LX
t
B(A)
n =
2,3
t
r
=
fOo.)
V
eiAtf (A)dA = fO 0
_00
has a corresponding representation Y
where
n
has the representation in terms of its spectral density X
Then
R
t
LX
=
t
i (ooe At B( A)f ( A)d lO
V
(BfO)
is called the Transfer Function of the fi Iter
IB( I
i
L
Writing
B(A)
A) e B( X), expresses the transfer function in terms of in polar form, B( A) = A (it may be a loss) and the the "gain" iB(A)1 i t produces at each frequency "phase shift B(~) • Because usually only real processes are considered, so
201
HiGHER DIMENSI01VAL SPEC1RAL l'ACJ"ORIZA'110N
that
X t
and
Y
will be real valued, one has
t
fOe A)
an even function.
An important class of linear filters are the convolution filters 00
Y
The kernel
k
t
= LX
00
(ooX(t-S)k(s)ds = J_ook(t-s)X(S)ds
t
is called the impulse response function. V
(Bf 0)
= k
*
Since
X
by the convolution theorem we have (BfO)( A) But
A
fOe A) = X (A)
=
A A
(K X )( A)
so we see that the transfer function
B( A)
transform of the impulse response function k B(A) = kA(A) important physical restriction (causality) is that k(s) = 0
for
s
<
is the Fourier A further
°
This means the output at time t depends only on the inputs at times s < t that is, only on the past and present of the process. Such filters a2~ then called realizable. By the usual identification of the Hardy space H on on the upper half plane with
L2(0,
00)
,realizable filters correspond to
A 2 impulse response functions k(s) with k in H + By Paley-Wiener theorems A one knows that then, recalling that k = B as shown above, ( _00
%lIB( A)ldA 1+1.2
>
(SKKKW)
The latter condition is, as we have indicated, sometimes called the Szego-Kolmogorov-Wiener-Krein spectral condition. Apparently and probably Krylov should be added as he arrived at a similar condition, although in a different context, in [7]. The factorization now comes about as follows. Among the impulse response functions k(s) , also sometimes called the filter by abuse of notation, which produce the same gain IB( A)I ,there is an optimal one kO(s) which is called the minimal delay filter or minimal delay impulse response. from any causal k( s) that produced the desired gain IB( A)I
= B( A)
a function in the Hardy space B( A)
2 H +
This is obtained by factoring k A ( A)
according to:
g( A)1jJ( A)
where 1jJ is an outer function and g is an inner function. Outer functions are characterized by their having no zeros in the upper half plane and by the fact that their absolute values satisfy a Jensen's type equality;
where the right hand side means the Poisson Integral from above 1jJ(0+) on the upper half plane. on the upper half plane and
P
K
of the boundary values
Inner functions satisfy
Ig(A)1
i
1
202
R. KENT GOODRICH and KARL GUSTAFSON
Ig(O+)1 =
a.e.
~
Outer functions are also characterized as all functions
--
V
in
2 H+
such that
2
sp{~ (>--s) Is ~ O} = L (0,00)
Note that
~V
of the given
2
is the inverse Fourier transform of the L (_00,00) ~
boundary values
in
FACTORIZATIONS IN HIGHER DIMENSIONS For the sake of simplicity we state our results for n = 2 parameter'dimensions. Analogous results hold for all n > 1 under suitable modifications. It is useful, both conceptually and practically, to now think of the parameter space, e.g., R2 the Euclidean plane, in a spatial sense rather than in a time sense. This corresponds naturally to studying approximation and digital filtering problems in several variables. There is a great deal of recent interest in signal processing and elsewhere in two and higher dimensional filtering problems. We cannot do justice to the wide and rapidly increasing literature on these problems and applications. As a sample see [8] and the references therein. As stated in [8] and elsewhere, the lack of a general factorization method has been a major obstacle to theory and application in higher dimensions. Remember that, as indicated above, the inner function must, among other duties, remove unwanted zeros. Remember also, as also mentioned above, the zeros of analytic functions of more than one variable may have a very complicated structure. Defini tion.
A function
~
in
2 L (R2)
is an outer function i f
Note the similarity of our definition to the one dimension characterization stated immediately above this section. Let R2
Let
v + U v
be a continuous unitary representation in a Hilbert space
We suppose
U
has a cyclic vector
, and
~O
, i . e. ,
be the projections of y ~
and sp{U(x,y)(o)ly Denote the range of any projection
P
by
H
~ d
R(P)
H
d
onto, respectively:
of
203
HIGHER DIMENSIONAL SPECTRAL FACTORIZATION
Definition:
v + U ( CPO) v
The mapping
is a regular process provided that:
for all
(s,t)
in
and
ns R(E s ) Theorem.
(Representation)
cyclic vector L2(R2)
CPo
Let
{oJ = nR(Ft) t v + U ( CPO) v
Then there exists a unitary mapping
V
of
H
with
H
onto
such that
VU V-I v
where
be a regular process on
R
v
R
=
v
is the regular representation of
R2
Moreover
is an outer function.
vie remark that the regular representation of
R2
is given on
2 L (R2)
by
(R/)(w) = few-v) The proof of the theorem may be found in [1]. The essential ingredient is the Stone-von Neumann Theorem. One uses the identities
and U(X,y)FtU(_X,_y) = F t +y The corresponding representations of W = JeixsdFx s
R1
given by
V t = JeiytdFy
and
satisfy the imprimitivity commutation relations
U
V U
(x,y) t
(-x,-y)
= e-ityV
and U
WU = e-is~ (x,y) s (-x,-y) s
The projection valued measure
p
corresponding (by Stone's Theorem) to
turns out to be quasi -invariant and hence
CPo'
U
, for Borel sets
v
A
defines a measure equivalent to Lebesgue measure. One then shows the family
{Uv'Vt'W } s
is irreducible.
That is,
the only
operators commuting with all of these are the scalar multiples of the identity. An application of the Stone-von Neumann Theorem completes the proof. We may now give our main factorization result on [1].
R. KENT GOODRICH and KARL GUSTAFSON
204
Theorem.
(Factorization)
representation such that function
g
on
Let
be any cyclic vector for the regular
~(
v +
is a regular process.
with \g(x,y)\ = 1
and an outer function
1jJ
on
R2
g
and
1jJ
a.e.
such that
Then there exists a
g( A) 1j!( A)
A)
are unique up to a scalar multiple of absolute value
The proof of the Factorization Theorem follows from the Representation Theorem by direct computation of the form of the isometry V One may weaken the notions of regular process and outer functions. In higher dimensions there are many versions according to the hyperplane and hyperquadrant commutativity requirements placed on the projections. As above, we restrict attention here to the two dimensional case. Definition.
Let
Hilbert space
H
v + U (
be a continuous unitary representation in the
with cyclic vector
v + U (
The mapping
is a weak
regular process provided that
F E
E F
s t
t
for all
s
(s,t)
in
R2
and
ns R(E ) = {O} = n R(F ) St t Definition.
A function
1jJ
in
sp{1jJV(v-w)\w = (x,y) ,
L2 (R2)
is a weak outer function i f
x> 0 ,
2
_00
< y < oo}
,
y ~ O} = L (- 00,00)
= L (O,OO)
2 x L (_OO,OO)
and
v
sp {1jJ (v-w) \w = (x,y) ,
_00
< x < 00
2
2
x L (0,00)
The above representation and factorization theorems hold if one replaces regular by weak regular and outer by weak outer. The proofs are similar to those outlined above; see [1]. Every regular process is a weak regular process, but not conversely. An example of a weak regular process that is not a regular process is given in [1]. The geometry of the support of the cyclic vector
As mentioned previously, we are presently investigating the
relationship of these factorization theorems to higher dimensional digital filtering.
HIGHER DTMENS/ON.4L SPECTRAL FACTORI/.Al'ION
REFERENCES: 1.
Goodrich, R.K. and Gustafson, K.E., Weighted trigonometric approximations and inner-outer functions on higher dimensional Euclidean spaces, J. of Approximation Theory, to appear.
2.
Goodrich, R.K., and Gustafson, K.E., Weighted trigonometric approximations in L2(Rn), in: Gustafson, K.E., and Reinhardt, W.P. (eds.), Quantum Mechanics in Mathematics, Chemistry, and Physics (Plenum Press, N.Y., 1981).
3.
Goodrich, R.K., and Gustafson, K.E., Regular representation and approximation, in: Sz. Nagy, B., and Bognar, J. (eds.), Froc. International Conf. on Functions, Series, Operators, Commemorating the 100th anniversary of the birthdays of Leopold Fejer and Frederick Riesz (North-Holland, Amsterdam, to appear).
4.
Gustafson, K., and Misra, B., Canonical commutation relations of quantum mechanics and stochastic regularity, Letters in Math. Phys. 1 (1976) 275-280.
5.
Kai1ath, T., A view of three decades of linear filtering theory, IEEE Trans. on Information Theory 20 (1974) 146-181.
6.
Dym, R., and McKean, H.P., Gaussian processes, function theory, and the inverse spectral problem (Academic Press, N.Y., 1976).
7.
Kry1ov, V.I., On functions regular in a half-plane, Mat. Sb 6 (1939) 95-138, A.M.S. Transl. (2) 32 (1963) 37-81.
8.
Mersereau, R.M., and Dudgeon, D.E., Two-dimensional digital filtering, Proc. IEEE 63 (1975) 610-622.
205
This page intentionally left blank
This Page Intentionally Left Blank
Spectral Theory of Differential Operators I. W. Knowles and R. T. Lewis leds.) © North·Holland Publishing Company. 1981
THE LIMIT POINT-LIMIT CIRCLE PROBLEM FOR NONLINEAR EQUATIONS John R. Graef and Paul W. Spikes Department of Mathematics Mississippi State University Mississippi State, Mississippi U. S.A.
An introduction to and discussion of the limit point - limit circle problem for nonlinear equations is given. In addition to a new nonlinear limit circle theorem, some relationships between the limit circle property and other asymptotic properties of solutions are indicated. INTRODUCTI ON In this paper we give some new results on the integrability of solutions of the perturbed second order nonlinear differential equation (a(t)x')' + q(t)f(x) = r(t,x) (I) similar in form to those known for the linear equation (a(t)x')' + q(t)x = O. (II) The Emden-Fowler equation x"+tOxY=O,y>l, (III) whi ch is a special case of .( I), wi 11 serve as our motivating model. H. Weyl [11 J in his classic paper on the subject classified equation (II) as being of the limitcircle type if every solution x(t) satisfies f OOx2 (w)dw < otherwise, (II) is said to be of the limit-point type. The reader can find an excellent discussion of the limit-point/limit-circle problem in the treatise of Dunford and Schwarz [4]. References to recent papers on this problem can be found in the monograph by Kauffman, Read and Zettl [7J. 00;
For the nonlinear equations (I) and (III) considerably less is known than for equation (II). In fact the only references seem to be the papers of Atkinson [lJ, Burlak [2J, Detki [3J, Graef [5J, Hallam [6J, Spikes [8-9J, Suyemoto and Waltman [lOJ and Wong [12J. While the other authors discuss limit-point criteria for unforced equations, only Graef [5J and Spikes [8-9J give limit-circle type criteria for forced equations. The exact form that intebrability results for solutions of (I) should take is not completely clear. The integrability results presented here will ensure that both x (w) f ( x ( w) ) dw < *) and fOOF(x(w))dw < where F(u) = f~f(w)dw (**) are satisfied, which is equavalent to showing that the solutions of equation (III), with y = 2n - 1 for some positive integer n, belong to L2n. Furthermore, (*) and (**) are consistent with the criteria of other authors for the limit-point/limitcircle problem for nonlinear equations.
f
00
00
00
207
(
j.R. GRAEF and P. W. SPIKES
208
A NONLINEAR LIMIT CIRCLE THEOREM We consider the equation (1)
(a(t)x')' + q(t)f(x) = r(t,x)
where a,q:[to,oo)c,R,f:Rc,R, and r:[to,oo)xR+R are continuous, a:q'€AClocLto'oo), a~q"€L21oc[tO'00), a(t)
>
0, q(t)
>
0, and xf(x) '" 0 for all x.
We define F(x)
=
J~f(w)dw and for any function 9 we let g(t)+ = max {g(t),O} and g(t)_ = max {-g(t),Ol. Also, the following assumptions will be utilized as needed. Assume that there exist positive constants A" A, C" C, and b s 2, nonnegative continuous functions hand k, and positive functions H,K€C'[tO'oo) such that H' (t)+ s (2-b)H(t)K(t)(a(t)q(t)/K(t) )~/2a(t)q(t), (K(t)H(t))~
s bCK2(t)H(t)(a(t)q(t)/K(t))~/4a(t)q(t),
'"
lr(t,x) I s
(5 )
2A,F(x) + 2A, 1"
h(t)(F(x))~
(6)
+ k(t),
Jt o[h(w)/(a(w)q(w) )2]dw L
00
(3)
(4)
CF(x) s xf(x) '" C F(x), l x2
(2)
(7)
< "',
J~ k(w)(K(w)/a(w)q(w))l"dw
<
(8)
"',
o
J;
H(w) k(w)( K(w) /a(w) q (w)) 12dw " "',
J~ [(a(w)q(w)/K(w))~/(a(w)q(w)/K(w))]dw
o
and
J;o
[l/H(w)K(w)]dw
Let
(9)
o
<
<
00
00.
(10)
(11 )
(2_b)K 5/ 2 (t) [(a(t)q(t)/K(t)),]2 /4a 3/ 2 (t)q5/2(t)
S(t)
+ [K 3/ 2 (t) (a(t)q(t)/K(t))' /a 1/2(t)q3/2(t)]' and assume that
J; oIS(w)ldw
<
J~ [IS(w)I/K(w)]dw
o
and
J~ oH(w) IS(w)ldw
(12 )
"',
<
"',
< "'.
(13 )
( I 4)
Remark. We point out that condition (6) is sufficient to ensure that all solutions of (1) exist on [to'oo). Theorem i.
If conditions (2) - (14) hold, then every solution x(t) of (1) satisfies
J~ x(w)f(x(w))dw
o
< '"
and
J; F(x(w))dw 0
<
00.
209
LIMIT POINT-LIMIT CIRCLE PROBLEM
The proof proceeds as follows.
Let s =J~o(q(U)/K(u)a(u)) 1/2du and let yes) = x(t).
Define R(t) = K3/2(t)(a(t)q(t)/K(t))'/4q3/2(t)al/2(t) and let· = d/ds. Then equation (1) becomes y + 2R(t)y + K(t)f(y) = K(t)r(t,y)/q(t) which is equavalent to the system y = z _ bR(t)y i: = (b-2)R(t)z + b[(2-b)R2(t) + R(t)Jy - K(t)f(y) + K(t)r(t, y)/q(t) Now let x(t) be a solution of (1) and define V and Wby V(s) = z2(s)/2 + K(t)F(y(s)) and W(s) = H(t)V(s). It can then be shown that there exists a positive constant t~ such that F(x(t)) :0; M/K(t)H(t) for t 2 to. The conclusion of the theorem then follows from conditions (4) and (11).
Remark. Due to the latitude in the choices of the functions Hand K, and the constants b, AI> A, C, and Cl ' Theorem 1 inc'ludes a number of known results. In particular it includes both Theorem 1 in [5] and Theorem 1 in [8]. It is also interesting to observe that when r(t,x) = 0 and f(x) = x, so that (1) becomes (II), then by taking b = 2, C = 1, hit) B k(t) =: 0, H(t) = (a(t)q(t))Y, and K(t) =: lour Theorem 1 reduces exactly to the well known limit-circle result given in [4]. RELATIONSHIP TO OTHER ASYMPTOTIC PROPERTIES We now state three results which give some relationships between the limit-circle property and other asymptotic properties of the solutions of (1). Theorem 2. Let the hypotheses of Theorem 1 hold. If H(t)K(t) is bounded below, then all solutions of (1) are bounded. If, in addition, H(t)K(t) + as t + and F(x) > 0 for x f 0, then all solutions of (1) tend to zero as t + 00
00
00
•
Remark. By suitable choices of the functions and constants in conditions (2) - (14), is is easy to see that Theorem 2 extends both Theorem 3 in [5] and Corollary 3 in [8].
For our next two results we need the condition f( x) fs bounded away from zero if xis bounded away from zero. Theorem 3. If (6) holds and f~o[(a(s)q(s))~/a(s)q(s)]dS
f ~ [( h(s) o
and
<
+ k( s ) ) / ( a ( s ) q (s ) ) y,] ds
( 16)
00,
<
(15)
00,
( 17)
F(x) + as Ixl + (18) then every solution x(t) of (1) is bounded. If in addition (15) is satisfied and x(t) is nonoscillatory solution satisfying 00
f~ x(w)f(x(w))dw
o
then x(t)
+
0 as t
+
00.
00,
<
00,
(19)
j,R, GRAEF and p, W, SPIKES
210
Theorem 4.
Let (6) and (15) hold,
f~ [(f~ [l/a(u)]du)h(s)j2ds o 0
<
00
and f~ (f~ [l/a(u)]du)k(s)ds o 0
<
00
If in addition (19)
and J~ F(x(s))ds
o
<
00
are satisfied for every solution sit) of (1), then every solution of (1) either oscillates or tends to zero as t ~ 00.
REFERENCES 1.
Atkinson, F.V., Nonlinear extensions of limit-point criteria, Math. Z. 130 (1973), 297-312.
2.
Burlak, J., On the non-existence of L2 -so1utions of nonlinear differential equations, Proc. Edinburgh Math. Soc. 14 (1965), 257-268. .
3.
Detki, J., The solvability of a certain second order nonlinear ordinary differential equation in LP(O,oo), Math. Balk. 4 (1974), 115-119.
4.
Dunford, N., and Schwartz, J.T., Linear Operators, Part II, Spectral Theory, (Interscience, New York, 1963).
5.
Graef, J. R., Limit ci rc 1e criteri a and related properti es for non 1 i near equations, J. Differential Equations 35 (1980), 319-338.
6.
Hallam, T.G., On the nonexistence of LP solutions of certain nonlinear differential equations, Glasgow Math. J. 8 (1967), 133-138.
7.
Kauffman, R.M., Read, T.T., and Zett1, A., The Deficiency Index Problem for Powers of Ordinary Differential Expressions, Lecture Notes in Mathematics No. 621, (Springer-Verlag, New York, 1977).
8.
Spikes, P.W., Criteria of Limit Circle type for nonlinear differential equations, SIAM J. Math. Anal. 10 (1979), 456-462.
9.
Spikes, P.W., On the integrability of solutions of perturbed nonlinear differential equations, Proc. Roy. Soc. Edinburgh Sect. A 77 (1977), 309-318.
10.
Suyemoto, L., and Waltman, P., Extension of a theorem of A. Winter, Proc. Amer. Math. Soc. 14 (1963), 970-971.
11.
Weyl, H., Uber gewohnliche Differentialgleichungen mit 5ingularitaten und die zugehorige Entwick1ung wi11kUr1icher Funktionen, Math. Ann. 68 (1910), 220269.
12.
Wong, J.S.W., Remark on a theorem of A. Wintner, Enseignement Math. (2) 13 (1967), 103-106.
Spectral Theory of Differential Operators I.W Knowles and R. T. Lewis leds.) © North-Holland Publishing Company, 1981
A MODEL SPECTML PROBLEM FOR TIlE LINEAR STABILITI OF NEARLY PARALLEL FLOWS H, IIerron* Department of Mathematics Howard University Washington, D.C 20059 1:;001
An analysis is made of a second order ordinary differential operator defined on (0, 00) with bOl.mdary conditions at O. This system models spectral problems which arise in the analysis of the "modified" Orr-Sommerfeld stability equation, when there is a transverse component as well as a streamwise component to the mainflow at infinity. The spectrum of the operator when defined on L [0, 00) is two-dimensional, consisting of an open, convex 2point spectrwn bOW1ued by the essential spectnnn which is a Jordan curve. The usual spectral resolution is therefore not possible in LZ[O, 00). The proper Hilbert space setting is determined and the spectral resolution is performed. INTRODUCTION The problem to be considered is to find ¢ELZ[O, 00) such that A¢,
(la)
0 < y < 00
where ¢ (0) O. To look for solutions in L2 [0, 00), we set ¢ = el3Y giving the characteristic equation (:<2+S+A=O so that
(lb)
1\ , 2 _-l±~ z .
Square integrable solutions exist when ReS I tive real parts if
<
IRe~1
Since A may be complex, set A = Al iiI - 4A - 4iA Z I
=
[,)(1 - 4AI)2 +
+
i;\Z'
0, ReS Z
<
< 1.
'Dms
16A~ei811/Z,
~"2 1 - 4\
Figure 1 211
0.
Both roots have nega(2)
212
where
IS0M H. HERRON
e
tan
-1
(-4,,/(1
4"1)).
[(1 - 4"1)
2
The inequality (2) is satisfied if and only if +
Z 1/4 e 16"2] cos 2' < 1.
Squaring both sides and noting that 2 cos [(1 - 4"1)
2
+
Z1
IS
1 + cos
=
2 1/2 16"Z] (cos
e+
e the inequality becomes
1) < Z
giving (see figure 1) 1 - 4"1
4"1)2
+ [(1 -
l6"zZ]1/2 < 2.
+
Simplifying, squaring again, and a final simplification gives (3)
Spectnnn
Resolvent Set
"
1
Figure 2
r
Thus for" in the interior of the parabola in figure 2, eigenfunctions are
<jJ
A
=
C /
1Y
e
BZy
),
A
t 1/2
-
eye- y / 2 ,
A =
1/4
The number of eigenvalues and eigenfunctions is uncountable. When the boundary value problem adjoint to (1) is studied it provides no relief. With the inner product (<jJ, ljJ) =
~
¢CYTljJ (y)dy,
(4)
00
o the adjoint is L*<jJ
=
Q.~ljJ
=-
= ~ljJ,
ljJ" + ljJ'
ljJ(O) = 0, ljJ ( L [0, 00).
2
If the same analysis is carried out assuming
l±~ 2
ljJ =
O
(Sa) (Sb)
eYY then solutions exist when (6)
A MODEL SPECTRAL PROBLEM ['OR LINEAR STABILITY
213
From (6) it is impossible that both ReYl < 0 and Rey z < 0 simultaneously. Thus there are no eigensolutions and hence no eigenvalues to (5). If moreover
IRe'; then both Rey 1 > 0 and Rey Z > O. same as (2).
1 -
4XI
< 1,
(7)
The values of ,\ for which (7) is true are the
The spectral resolution of the operator L in (1) is therefore in doubt. The usual spectral resolution is for a countable number of eigenfunctions, generalized eigenfunctions and/or eigen-elements associated with a continuous spectrum. In a different context Shinbrot [1] encountered an integral operator in LZ whose point spectrum was uncountable and two-dimensional. Shinbrot proved that there existed for the operator a sequence, contained in the set of all eigenfunctions, which was complete in the orthogonal complement of the union of the null space of the adjoint of the operator with the null spaces of all positive powers of the adjoint operator. The means for resolving the difficulty here is different. Here, we modify the problem so that it is defined on another Hilbert space H ~ Lz' and the spectral resolution performed in H. First a number of preliminary definitions and lemmas are given.
SPECTRAL SETS Definition 1 [Z, p. 1187]. H such that dmn L c H.
Let L be a closed operator in a complex Hilbert space
The resolvent set of L is the set of complex numbers ,\ such that (L - A)-l exists, is bounded and defined on all of H. This set is denoted by peL). If A¢p, then Aso(L), the spectrum of L. The point spectrum ° (L) is the set of eigenvalues ,\ p for which L - A is not one-to-one and hence not invertible. The continuous spectrum, 0c(L) is the set of complex numbers A for which L - A is one-to-one and has a dense range rng(L - A) f H, but the closure of the range rng (L - A) = H. The residual spectrum is the set 0r(L) of complex numbers A for which L - A is one-toone and has a range not dense in H. Thus the sets 0p(L) , 0c(L) and 0r(L) are mutually disjoint and () (L)
=
°p (L) u e ° (L) ur () (L).
Definition 2 [2, p. 1393]. Let L be as in definition 1. The essential spectrum of L is the set of complex numbers A such that rng(L - A) is not closed. It is denoted by 0e(L). Lemma 1.
Let L be as in the previous definitions.
Then 0e(L)
~
0c(L).
Lemma Z. Suppose L in lemma I_is also densely defined. Then L has a unique adjoint L*. I f A s °r (L), then A s °p (L*). If II s °p (L*) , then jj s °p (L). Corollary. If L o/L) is empty.
=
L*, then
°
r
(L) is empty.
°
If L f L* and p (L*) is empty, then
More general versions of these lemmas are proved elsewhere [3]. When L is defined by a constant coefficient differential operator ~ with boundary conditions, as in the model problem, even more can be said. Rota [4] has proved that for constant coefficient differential operators ~ of any order, defined on [0, 00), the essential spectrum 0e(~) is determined by the set of points where
IS0M H. HERRON
2J4
9-¢ = Acjl has solutions of the fonn ¢ = e iwy , w real. Then mg (9- - A) is not closed. He proved furthennore that the boundary conditions at 0 do not change the essential spectnnll. It is sensible then to speak of 0e(L) and 0e(L) = 0e(9-). r~ldberg [5, p. 1631 proved that (for constant coefficient operators 9-) when A E °e(9-) , then rug (9- - A) is a proper dense subspace of H which means
AEOe(£)
¢=¢-AsO (£)· c
Thus we conclude on the basis of lerrulla 2 and its corollary that since (3) represents p (L) and ° p (L*) is empty then r (L) is empty. With the results of Rota and Goldberg, since Land L* have constant coefficients, 0e(L*) = 0e(L) is the parabola in figure 2, 0e(L) = 0c(L) = 0c(L*) and (7) represents 01' (L*) .
°
°
SPECTRAL RESOLUTION The spectral resolution of a spectral operator can be performed, based on the following theorem. Theorem [6, p. 271J. Let the operator Ll in the Hilbert space H have its spectnnn + o(L l ) in a Jordan curve f. Denote the two "edges" of f by f and f. Let D and D* be dense subspaces of H with the following three properties: i)
For fl
E
D, £2
E
V*, there is a constm1t K, depending on £1 and f2 such that < K,
(8)
for A ¢ f, dist CA, f) suffici ently small. ii)
For each £1
E
D, f2
D*, the linuts
E
+
R (A O,f l '£2) R-(AO,fl,f Z) exist for each point AD iii)
t:
f,
=
lim ' A-;'A~
the limit heing taken in a non-tangential manner.
There is a constant C depending only on L , such that l
f IR+(A,fl'fz) ° (L l )
- R-(A,fl ,f2) Ids < Cllflll IIf211,
(9)
for f l , £2 E H, s being the arclength on f. Then Ll is a scalar spectral operator wllose spectral resolution is given by the fonnula
= -
Z;i
J
[R+(A,fl,f z)
-
R-(A,f l ,f 2)]dA
(10)
e
where E(e) is the spectral projection defined on the fmnily of Borel sets e ~ o(L ). l This theorem does not apply to L as previously defined because of the structure of its spectrum. However we can define the appropriate operator Ll . Consider a nonhomogeneous version of (1). Suppose
A MODEL SPC;CTRAL I'IWIlLEAI FOR LINEAR ST.1BlLITY
O
Q.o¢ = - ¢" - ¢' - A¢ = f,
with where [
L [0, co).
E
0,
¢CO) Set ¢(y)
z
-\)J" + with
=
ella) (llb)
e -y/24J(y) in (lla) so that
(j - AN \)J(O)
The solution will be
215
eY/ 2 f
=
(lZa)
0.
=
(lZb)
i g(y,~;II)e(/Z[(~)d(,
(l3a)
e -rly-E:I - e -r(y+() 2r
(13b)
00
\)J(y)
o
where and r
g(y,(;A) =
=
-Jj -
II is the positive square root.
The Green's function (l3b) satisfies
(lZa) with the right side replaced hy 6 (y - () the usual Dirac ftmction. is the kernel of a bOlll1ded integral operator: L [0, 00) -+ L [0, 00), for Z Z A ¢ {v E IR Iv ;
TIllis g
j}.
Now define
00
(L
where cbnn L]
- A) -If(y)
l
{¢
E
=
J g(y ,(;A)e - (y-()/2 f (S)d(
°
HI¢, ¢' absolutely continuous, "'O¢ L ¢ 1
=
¢
Q()ctl ,
f
H},
E
Lz[O,
C14b) (l4c)
dnm L , l
E
H = {¢ley / 2¢
E
(14a)
co)},
(lSa)
if>
and
<¢,\)J>
is the inner product on
H.
Cy)\)J(y)cYdy
(lSb)
°
TIle application of the spectral theorem to Ll now follows. The first hypothesis z is satisfied by taking V = V* = H n B where B = {¢ IeY/ ¢ E Ll [0, 00) L Then for f l , f2 E V, II E peLl), OO
I
l
- A)-lfl>1
I jOOeY/Zfz(y)dY
o
J
g(y,(;A)e V2 f (()d(1 l
C16a)
0
(16b) by Schwarz's inequality where
r'e °
Y
IfCy)
sup (II (L
fEV
l
2 1
dy
- A) -lfll/II£II).
(16c) (16d)
IS0M H. HERRON
216
What is needed next is that g(y,~;A) is bounded in A. From (13b), (14a) the only singularities of gCy,~;A) in A can occur where rCA) = O. Now lim gCy,I;;A) = r (A )-+0 [-\y - 1;\ + (y + 1;)]/2, which is finite. Thus condition (i) holds. It follows that Op(L ) is empty since g(y,I;;A) has no poles. Hence O(L l ) = crc(L l ) = oe(L l ). l We take r = cre(L ) = {A € !RIA ~ This half-line is given parametrically by l
i}.
2
(17) 1/4, The verification of condition (ii) follows with the identification; if AO € r, then from (7), AO = w~ + 1/4 for some nonnegative wOo Next take lim+ rCA) = r(A~) = - iwO and lim_ rCA) = r(A~) = iwO in (13b). Thus the limits A-+AO r-+AO exist since the terms involving rCA) are bounded and the conditions on fl and f2 ensure the convergence of the integrals. A = w +
Condition (iii) is the most involved. Let the arclength s = A so that ds = dA = 2wdw. It tallows from (13b) and the last paragraph that
j( IR+(A,fl ,f2) - R-(A,fl ,f2) Ids r
=
fo f
=
co
00
I
f'2CYTdy
0
<
C
f
00
y 2 e /
[2
0
2
JoIF2 (w) I dw 00
by Schwarz's inequality, where
F.J Cw) = '1TI !f
Joosinwy F. (y)dy is the sine transform
o
J
of F.(y) = ey!2f . Cy), j = 1, 2, since F. (y) E L2 [O, (0). The inequality (9) J J J follows by application of Parseval's formula for sine transforms. Thus the operator Ll has a spectral resolution. tegration.
Figure 3 denotes the path of in-
A
l---------------~~
'4
Figure 3 CONCLUDING REMARKS The spectral problem suggested by (la, b) models the linearized stability equations in Ci) the case of the asymptotic suction profile and in Cii) the case of a
217
.1 MODHL SPECTRAL PROBLEM FOR LlNliAR STAI3ILITY
homogeneous symmetric jet, when there is "inflow from infinity" [::']. The class of adnlissible disturbances is restricted in this theory. It would be interesting t? knO\~ if some subset of the L2 cigenfLmctions could be used to resolve the sltuatlon.
REFERENCES: [11
M. Shinbrot, Eigenfunction Lx:pansions Associated with an Integral Operator, Trans. Am. Mel.th. Soc. Dl (1965) pp. 143-156.
[2]
N. Thmford and .LT. Schwartz, Linear Operators I, II, Irr,(Interscience, New York, 1958, 1963, 1971).
[3]
I.II. Herron, Expansion Problems in the Linear Stability of Boundary Layer Flows (In preparation) .
[41
G.C. Rota, Lxtension Theory of Differential Operators, Comm. Pure App. (1958) pp. 23-65.
II
~!ath.
[5]
S. Goldberg, Unbounded Linear Operators,(McGraw-Hill, New York, 1966).
[6]
N. Thm[ord, II Survey of the Theory of Spectral Operators, Bull. Am. Hath. Soc. 64 (1958) pp. 217-274.
*This work has been supported through a contract with the Office of Naval Research (ONR).
This page intentionally left blank
This Page Intentionally Left Blank
Spectral Theory of Differential Operators I.W Knowles and R. T. Lewis (eds.) ©North-Holland Publishing Company, 1981
TITCHMARSH-WEYL THEORY FOR HAMILTONIAN SYSTEMS
Don Hinton Mathematics Department University of Tennessee Knoxville, TN 37916 USA
Ken Shaw Mathematics Department Virginia Polytechnic Institute and State University Blacksburg, VA 24061 USA
For linear Hamiltonian systems, a Titchmarsh-Heyl matrix M(A) function is defined. The systems formulation used by Atkinson is employed under a limit point hypothesis. A theory analogous to the second order scalar case of Chaudhuri and Everitt is developed. Characterizations are given for the resolvent set, point spectrum, continuous spectrum and point-continuous spectrum. An invariance with respect to boundary conditions is established for certain parts of the spectrum.
PART I.
THE M(A) FUNCTION
We consider the 2n
-7
where y is a 2n 2n
x
x
x
2n Hamiltonian system
J
y'
=
[H(x) + B(x)
]y,
a < x < b*
<
(1.1)
co
1 vector, A is a complex parameter, and A and B are continuous
2n complex matrix functions; further A(x)
A*(x)
>
0,
Bi, (x) ,
B(x)
where In is the n x n identity matrix.
This formulation includes the symmetric
ordinary differential expressions of order 2n (cf. [14]) and also the Dirac systems of [12]. A solution
y of
(1.1) is said to be of integrable square if J ->-
2
denote this by y E LA'
219
bole -+
a
y* AY
<
00, and we
DON B. HINTON and K. SHAW
220
The classic Titchmarsh-Weyl theory constructs, for the second-order differential equation -(py')' + qy half-planes.
Ay, an analytic function m defined in the upper and lower
=
The function m is unique under a limit-point hypothesis and is in-
strumental in the investigation of the integrable-square solutions and the spectrum of the differential operator.
We refer to the excellent survey article by
Everitt and Bennewitz [6] for a discussion of the function m. This paper outlines a Titchmarsh-Weyl theory for the system (1.1).
An
n x n
matrix-valued analytic function M(A) is obtained by which L! solutions of (1.1) are constructed, and the spectrum of the Hilbert space operator induced by (1.1) is characterized. The mCA) functions have been developed already for symmetric scalar differential expressions by Everitt [3,4,5] (see also the survey papers by Everitt and Kumar
[7,8]).
The text of Levitan and Sargsjan [12] constructs meA) functions for
Dirac systems.
The results obtained here are in agreement with these works.
For
relating the spectrum to the meA) function, we take as our model the fundamental paper by Chaudhuri and Everitt [2].
The results of Part I extend some of the
work in [9]; those of Part II show how the results of [10] may be formulated for more general boundary conditions than considered in [10]. Following Atkinson [1, Chap. 9], we assume the definiteness condition, i.e., J
d
-+
y'~
c
..,.
Ay
> 0
-+
if Y is a nontrivial solution of (1.1) and a
<
(1.2)
c
<
d
<
b*.
-+
E L2)
We assume also the
"limit point" hypothesis lim
-+
-+
y* (x) J z (x)
o
z
(1. 3)
A
x-+b;'
for all solutions
y of
(1.1) and; of J ;,
=
(VA + B);, i.e., A and V may be dif-
ferent.
In case (1.1) is the matrix formulation of a symmetric scalar equation,
the term
y*
J ~ in (1.3) is the usual Lagrange bilinear form, and (1.3) is then
equivalent to the associated scalar operator being of limit-point type [13, p. 19].
Define 2""
-+
SeA) = {y E LA: y is a solution of (1.1)}. Applying Theorem 9.11.1, p. 295, of [1], a calculation shows that for 1m AID, dim SeA) ~ n and dim SCI) > n.
It has been shown by Kogan and Rofe-Beketov [11]
that dim SeA) is constant in the upper and lower half-planes. LEMMA 1.1.
16
(1. 3)
hold;."
.the.n dim S (A)
dim SeA)
n
SOfl.
1m A 1
o.
221
TITCHAIARSH-IVEYL THEOR Y
PROOF.
that ~*(a) J di~ts
Sin~e J is non-singu-
If not, then for some A, dim SeA) + dim sCi) > 2n.
lar, then dim J S (i)
1" (a)
=
Hen~e there is a ~ E S (A) and a ~ E S (i) su~h
dim S (i) .
:f o.
A differentiation shows
;* J;
is constant; this contra-
(1.3) and proves the lemma.
A differentiation establishes that if ~ is a solution of (1.1), then (1. 4)
existen~e
To prove the
~lassi~al
of the
meA) function, a regular boundary value
problem is associated with the differential operator.
We follow the same method
and associate with (1.1) the eigenvalue problem (1.5)
o where the n
n matrices aI' a , 13 , 13 satisfy: 2 1 2
x
(1. 6)
The eigenvalue problem (1.5) may be put in the parametric form of the text of Atkinson, i. e. ,
J~' where
M~'JM =
N'~JN
(AA + B)~,
~(a)
and Mu
0 implies u
Nu
=
M
= (
~
a --a
=
,', 2
*
),
Ya'
where J
Green's matrix
Y'a
= (AA
+
whi~h
B)Ya ,
Nv
=
for some v f 0
c; ~ ).
0, by choosing, N
l
The above problem is symmetric and has no asso~iated
~(b)
Mv,
-i3'~
1
~omplex
eigenvalues.
may be constructed.
Thus there is an
Take as a fundamental matrix
and
Ya (a) The conditions (1.6) ensure that Ea is non-singular. loss of generality that
..,.
We partition the matrix Y into n a
x
n blocks by
We further assume without
222
DON B. lIINH)N and K. SHAlt'
Y (x,A) a
~(X'A»)
__ (S(X,A) .. S(x,A)
¢ (x, A)
and also use the notation
3(X,A)
=
SCX,A») ( SCx,A) A
,
~(x,A)
(¢(X,A») =
A
Ijl(X,A)
These are the matrix analogues of the scalar functions (cf.
e, ¢
used by Titchmarsh
[6]).
The Green's matrix of (1.5) [1, p. 265] is given in terms of a characteristic function F = FMN(b,A) [1, p. 269].
In our notation, F satisfies (for A not an
eigenvalue) Ea
l
(FJ
1 + -2
I)E
(1. 7)
ex
Some calculation reduces (1.7) to (1. 8) where (1. 9)
A property of F that is crucial to our development is [1, p. 289]:
A slight modification of the proof of [1] shows F is uniformly bounded in band A for A restricted to a compact set containing no real numbers. Following now the scalar case we compute a 2n x n matrix solution ~b of (1.1) of the form ->-
(8
+
->-
(1.10)
¢ C
->-
which satisfies the right-hand boundary condition of (1.5), i.e., [Sl,S2]'I'b(b) =0. Substitution of (1.10) into this boundary condition yields that C = Ma(b,A) where Ma(b,A) is given by (1.9). LEMMA 1. 3.
We now investigate the behavior of Ma(b,A) as b
->-
b 1,.
223
TJTCIIM.JRSIl-II'EYL THEORY
PROOF.
From the relations
[ 8~] 0,
n,
=
-8~
we conclude that for some matrix
r,
->
~b(b)
[8 ,-8 1*r. Z l
A short calculation now
yields the conclusion. -+
Replacing y by
-+ ~b
in (1.4) yields (1.11)
16
THEOREM 1. (i)
dim SeA) = dim SeA) = n bOJL 1m A .; 0, then bO!L all A wilh Im(A) f 0,
M (A) =: lim;, Ma (b, A) eJe-ud;, clf1d ;;, ~ndependent a b+b Ma (A) ;;, analyulC and IW6 !Lank n;
0
b
8
1
and 8 ; further Z
(ii) (iii)
(iv) PROOF.
the m~x M CA) - M'~(A)/Im A ;;, pMili.ve deMnLte; M (A)
a
a M'~(i).
a
a
From (1.8) and Lemma 1.2 we conclude that for b
-+ b* as n + 00, {Ma(bn,A)} n C as n -+ 00 through a subsel+ a n ""* quence. Letting b = b in (1.11) and defining IjIl = Y [I ,C*l'~, we conclude that n a n 1 the columns of 11 are in L~. The matrix In in the definition of ~l implies the
has a convergent subsequence.
Suppose M (b ,A)
-+
-+
columns of IjIl are linearly independent; hence they are a basis of SeA).
If C
is
2
the sequential limit of another sequence {Ma(dn,A)}, then similar reasoning may
- Ya[I ,C;11,. Since the columns of both 11 and n Z we have for some n x n matrix C , be applied t01
>P2
span sO),
3
hence C = In and C = C . This establishes the existence of the limit. Similar l 2 3 reasoning shows the limit is independent of 8 and 8 . The analyticity of Ma(A) Z 1 follows from the uniform boundedness properties of F. The relation (1.11) yields a proof of (iii).
The relation (iii) implies the rank of Ma(A) is n.
To establish (iv), note that the choices (Sl
I
n
)
in
(1.9) yield by (i), M
a
CA)
lim hp (b ,A) -1 b+b;'
e (b, A) }
lim { - $ (b , A) -1 b-+b*
e (b, A) }.
(LIZ)
DON B. HINTON and K. SHAW
224
A differentiation shows y1'(b,~) JY (b,I) B y1'(a,~) JY (a,I) a
a
a
a
J.
Reversing the
order of products gives that
The upper left-hand corner of this equation is
o
= e(b,I) ¢* (b,A) - ¢(b,i)
e;'
(b,A).
This relation and (1.12) completes the proof of (iv).
PART II.
SPECTRAL THEORY
The inner product defined by ->-->-
generates only a seminorm
f
b*+ ->f>~(t) A(t) get) dt a
II ->-f IIA =
-)- 1> 11/2 ,unless A(x) is invertible. [
How-
ever, to write the second order equation -(p(x)y')' + q(x)y = AW(X)y, w(x) > 0, in the form (1.1) we take ([1, p. 253])
For the Dirac systems of [12] we note that A(x) = I
2n
.
To allow for these cases
we will assume henceforth that A(x) has the form A(x) __ [A10 (x) where A1 is r x r and invertible, r
2n.
Letting
{E)~Tg E L!}, we note that Er L! is a Hilbert space under the inner
2
and Er LA product <
<
,
>.
We wish to view the boundary problem given by (1.1) and [al,a21 yea) operator equation.
o
as an
Note that (1.1) may be written
0]o
-+-
-r
-+
[Jy' - By] =AE y. r
This suggests defining an operator T, with domain D(T), as follows: and only i f (i) y(x) E L!;
y E D(T) if
(ii) y(x) is locally absolutely continuous on
[a,b'~);
225
TlTCHMARSH-TI'LYL THLOR Y
(iii) [a ,a Jy(a) l 2 in (1.5);
0, where [a ,a J refers to the boundary conditions introduced l 2
=
-+
(iv)
-+
[Jy' - ByJ r-: ErL
2 A
0;
; (v)
and
A~ 1 (x) T y(x) =
[
0
In [9J we prove uniqueness of L2 solutions of (1.1) for Im(A) # O.
Stated in
A
2 operator-theoretic terms, T - AE : D (T) -+ E L is one-to-one and onto for r r A
Im(\) # O.
A consequence of the proof is that (2.1)
for rm(\) #
o.
If for some real or complex A, (T - AE )-1 exists as a bounded operator defined 2 r 1 on all of Er LA' then we call RA (T) = (T - AEr) - the fLv>ofvent: opeJta;(:ofL corresponding to A.
oet oE T.
The set peT) of all such A is called the fLv>ofvent:
is defined to be the complement of peT) in the set of complex numbers. of isolated points of aCT) is called the peT). E (T)
po~nt:-opeQtfLum
a (T) - P (T) is called the
V>Mnt:.i.af
The set
of T and is denoted by
We will show that the elements of peT) are eigenvalues of T. =
Of
The opeQtfLum aCT) of T
course, (2.1) implies that A E peT) whenever rm(A) # O.
opeQtfLum of T.
The set
The subset PC (T)
C
E (T)
consisting of eigenvalues in the essential spectrum, those \ E E (T) for which (1.1) has a nontrivial solution in D(T), is called the set C(T)
=
E(T) - PC(T) is called the
Qont~nuouo
pobu:-Qo~nuouo
opeQtfLum.
The
opeQtfLum.
The following theorem generalizes the fundamental result of Chaudhuri and Everitt [2] for the scalar second order case.
THEOREM 2.
Let
1,0
be a Qompfex numbefL.
(i)
Ao E p (T)
=
Ma (\) ~ anafytiQ
(ii)
1,0 E peT)
=
Met (\) hM
a
o~mpfe
Then
at
Ao;
pofe at 1,0;
(iii)
(iv)
\0 E PC(T)
anafyuQ
at
=
lim \)-+0
1,0'
not
226
DON B. HINTON and K. SHAW
We will now outline the proof of Theorem 2 and give formulas for the resolvent Further details may be found in [10].
operators in parts (i) and (ii).
In [9] we computed the Green's matrix for the boundary problem (1.1) and used it to establish (2.1).
Let
~(t,A)
=
S(t,A) +
~(t,A) Ma(A) be the unique L! solution
of (1.1) established in the proof of Theorem 1.
Let us recall that the Green's
matrix has the form x < t,
(2.2)
K(X,t,A) x > t,
K by
and define an operator
f
b'~
-+
K(X,t,A) g (t) dt
(2.3)
a
for
gE
L2 and 1m (A) A
+ O.
We note that (2.2) can be written, for 1m (\)
if
a
K(X,t,A)
if
(X,A) [ 0 I n
If we let
y=
(t,);');',
0
as
x < t,
a (2.4)
~ * I ] -+ Y (t,A) , M (\) a a
(x, \) [ 0
a
o ] if M (A) a
+ 0,
x > t.
K(X,A,g) and use (2.4), then direct differentiation gives (2.5)
-+
since Y is a fundamental solution matrix. By an identity in [1, p. 269], the a right side of (2.5) reduces to A(x) g (x), or in other words T = g. These steps
y
are permissable even if \
=
AO is real, provided M(A) is analytic at AO.
However,
we have to show that (2.3) is actually defined for real \0; i.e., we need to establish that the columns of ~(X,AO)
8(X,A ) + ~(X,AO) Ma(AO) belong to L!. O
For
this we rely on the identity fb a
1 '
~;'(t,A) A(t) lJI(t,A)dt
which is the limiting case of (1.11). analytic at AO' recall that ~
1m Ma(A) 1m (A) f 0,
1m A Put A
e + $ Ma(\)
=
(2.6)
AO + iv in (2.6), suppose Ma(A) is
and let v
-+
O.
The right side of
(2.6) approaches M' (A )' and so an appeal to the Lebesgue convergence theorem O -)b~'( -)lJI(t,A ) A(t) lJI(t,AO)dt < ro Standard operator-theoretic arguments may O now be used to identify (2.3) with the resolvent operator. Hence AO E peT) whenyields J a
ever AO is a regular point of Ma(A).
227
'111'CHM.. } RSH-It'rYL THEOR)'
The basis for the other direction in (i) is the identity, valid for 1m (A) M CA)
(A - i) f
M (i) C!
C!
+
(A
+ 1) f b""'
2
b . .'~
.-+....
-1-
,¥"(t,i) A(t) 'I'(t,i)dt
a
-
+ 0,
+
K(t,A, '!' (',i»
(2.7)
* A(t) --+-'¥(t,i)dt,
a
whose derivation may be found in [10].
If we start with AO E peT) then a separate
argument, based on the fact that K(t,A,') is known to be the resolvent operator of T for 1m (A)
+ 0,
establishes that the right side of (2.7) is analytic in a
neighborhood of AO (see [10]).
Thus (2.7) gives the analytic continuation of
Ma(A) to AO' Concerning (ii), to say that Ma(A) has a simple pole at AO means that M (A)
a
°-1 (A
(2.8)
in a neighborhood of AO' that ok
~
From the symmetry relation of Theorem l(iv), we know
0; the matrices ok are size n
point of the spectrum o(T).
x
n.
From part (i), AO is an isolated
This shows that AO E peT).
On the other hand, if
Ao E peT) then part (i) ensures that AO is an isolated singularity of Ma'
From
Theorem l(iii) we know that the diagonal entries of Ma belong to the PickNevanlinna class; i.e., {1m (A)} • {1m (Ma)kk(A)} > O.
Thus the diagonal entries
can have simple poles at most at any isolated singularity. M (A) - M (i) a
a
=
(Ie - i)
Now the identity
Jb1'~'~(t,I) A(t) I¥(t,i)dt a
may be used to bound the off-diagonal entries by expressions involving the (Ma)kk'
Indeed, the Cauchy-Schwarz inequality gives
If we multiply this by v, where A = AO + iv, and note that iv(M ) . . (A Ci.
remains bounded as v
+
]]
O
+ iv) i
0, then we may conclude the same about (M )'k' i.e., all Ci.
]
entries have simple poles at most. To identify eigenfunctions at points AO E peT) we again use (2.6). this time yields
Letting v
+
0
228
DON B. HINTON and K. SHA TV
from which it follows that the columns of $(t,AO)o_l belong to L!. columns are eigenfunctions.
Thus the
The number of linearly independent eigenfunctions
clearly equals the rank of the residue 0_1' If A = AO E peT), the nonhomogeneous problem (2.9) 2
->-
where g E LA' obviously does not have unique solutions, due to the presence of eigenfunctions. if
g is
Nevertheless, it can be proved that (2.9) can be solved uniquely
orthogonal to the manifold generated by {$(o,AO)o_l}'
In terms of oper-
ator theory, the operator T admits a "reduced" resolvent defined on the orthogonal complement ErL! 8{
We now describe the reduced resolvent. -t
Thus let AO E P(T), A = AO + iv and let
t
2
->-
E Er LA 9{
-t If [R (T)t]
A denotes the resolvent corresponding to A, then by (2.2) and (2.3) we have ->-
[Ric (T) f] (x)
[8(X,A) + $(X,A) M (A)] IX $*(t,i) A(t) f(t) dt a
a
(2.10)
+ 1;(X,A) Jb* l0(t,i) + ~(t,i) M (i)]* A(t) f(t) dt. x
Subtracting off the term 0
=
a
<¢(o,AO)o_l' f>, (2.10) becomes [R\(T)f](x) ->-
[8(X,A) + ¢(x,\)M (\)] JX $1«t,i) A(t) f(t)dt a a
+ $(x,\) I
b* X
-
¢(X,AO)o_l
[8(t,i) + $(t,i)M (i)l'~A(t) f(t) dt a
Putting Ma(\)
MaCA) -
-1
l (\ - \0) be expanded further to [R\ (T)f] (x) = =
CJ_
-
and
\ - \0
$(x,A O) \ - Ao
IX $1«t,\0) A(t) f(t)dt a
*
II\~(t'\O)o_l«A(t)f(t)dt. x
$(t,\) - $(t,A )' (2010) may O
0(x,\) IX $* (t,i) A(t) f(t)dt + 1;(x,A) M (\) IX 1;* (t,i) A(t) f(t)dt a
a
Letting \
->-
a
\0 in the third line above formally gives the expression (where a
subscript \ denotes partial differentiation)
229
Arguing as in [2] we can show that this integral actually converges and that the limit as A + AO may be taken as indicated.
Similarly, multiplying and dividing
by A - AO in the last line of (2.11) and using standard Legesgue theory arguments gives for the last integral -~A (x,A ) f~* [;t"(t,AO)O_l]* A(t) fet). O The limit of the second line of (2.11) is
-7-
Thus letting A + AO in (2.11) and using the orthogonality condition on f leads to the expression -+
-+
X
-+.1..
7-
[8(x,A ) + (x,AO)oO] fa V(t,A ) A(t) f(t) dt O O
+ ;t"(x,AO)o_l
fX
a
¢~(t,AO) A(t) f(t) dt
(2.12 )
which is formally the correct expression for the reduced resolvent.
That (2.12)
may rigorously be identified with the resolvent operator follows from standard operator theory arguments.
We omit the details.
If liw vM (AO + iv) = S ~ 0 and M (A) v+u a a is not analytic at AO' then we cannot have AO E peT) as otherwise
We briefly discuss (iv) and (iii). -1
is(\ - Ao)
lim vM (AO + iv) = O. We cannot have AO E peT) for in that case we would have v+O a -1 S = -io_ and analyticity of Ma(A) - 0_1 (A - AO) Thus AO E E(T) and we have l 2 only to demonstrate existence of eigenvalues. That <1>(. ,AO)SE LA serves this purpose follows "from a modification of the analogous part in [2]. The proof that AO
~
Thus AO E PC(T).
PC(T) implies the conclusion of part (iv) also follows as in
[2], and we omit these details. As for part (iii), let AO E C(T). exhaustive subsets of E(T).
By definition C(T) and PC(T) are exclusive and
Hence Ma(A) cannot be analytic at AO.
In fact,
Ma (A) - is(A - AO)-l cannot be analytic at AO for any S as otherwise the singularity would be isolated.
Therefore the condition VMa(A
O
+ iv)
+
S
~
0 is ruled
out by (iv). If M (A) is not analytic at AO' then AO ¢ peT). If lim vM (AO + iv) = 0 then a v+O a Ma(A) cannot have a pole at AO since the value of the limit is the residue times
230
-i.
DON B. HINTON and K. SHA W
Thus AO E PC(T) U C(T).
The limit condition vMa(A
O
+ iv)
0 excludes PC(T)
+
by part (iv). We close with a few remarks on invariance of the spectrum.
First, it is possible
to compare Ma(A) functions arising from different choices of the matrices aI' a of (1.5).
2
It is simplest to do this through the special choice a
and the corresponding function which we denote simplY by M(A).
= In' a = 0 l 2 Let us write
o/a(X,A) and o/(x,A) for the corresponding unique L! solutions constructed in Theorem 1.
Invoking the limit-point hypothesis, the number of independent L!
solutions requires that 'r(X,A)
0/a (x,A)
C , where C is an n
x
n nonsingular
*
*
For x = a we obtain I [a~ - a; Ma (\)]C and M(A) = [a + a l Ma CA) ]C. 2 The first of these equations implies that [a * - a * Ma(A)] is invertible for all 2 l A, and the second therefore gives
matrix.
M(\)
=
[a* + a* M (A)][a* - a* M (\)]-1. 2
1
Ci.
1
2
a
This is analogous to a linking formula of Chaudhuri and Everitt [2]. Following the argument of [2], we may now establish the following invariance principles for spectra of operators T
Ci.
choices of matrices aI' at x
=
Ci.
and T
y
arising from different admissable
and Y , Y which determine the boundary condition 2 l 2
a: p (T )
a
U P (T )
a
p (T )
Y
U P (T ); Y
REFERENCES [1]
F. V. Atkinson, "Discrete and Continuous Boundary Problems," Academic Press, New York, 1964.
[2]
J. Chaudhuri and W. N. Everitt, On the spectrum of ordinary second order
differential operators, Proc. Royal Society Edinburgh 68A (1967-68), 95-119. [3]
W. N. Everitt, Fourth order singular differential equations, Math. Ann., 149 (1963), 320-340.
[4)
W. N. Everitt, Singular differential equations, I; the even order case, Math. Ann., 156 (1964), 9-24.
[5]
W. N. Everitt, Integrable-square, analytic solutions of odd-order, formally symmetric, ordinary differential equations, Proc. London Math. Soc. (3), 25 (1972), 156-182.
(6)
W. N. Everitt and C. Bennewitz, Some remarks on the Titchmarsh-Weyl m-coefficient, in: Tribute to Ake Pleijel, Department of Mathematics, University of Uppsala, Sweden, 1980.
TITCIIM-IRSlllt'L:YL THEOR Y
231
[7]
W. N. Everitt and K. Kumar, On the Titchmarsh-Weyl theory of ordinary symmetric differential expressions I: The general theory, Nieuw Archief Voor Wiskunde (3), 24 (1976), 1-48.
[8]
W. N. Everitt and K. Kumar, On the Titchmarsh-Weyl theory of ordinary symmetric differential expressions II: The odd-order case, Nieuw Archief Voor Wiskunde (3), 24 (1976), 109-145.
[9]
D. B. Hinton and J. K. Shaw, On Titchmarsh-Weyl M(A)-functions for linear Hamiltonian systems, J. Diff. Eqs., to appear.
[10]
D. B. Hinton and J. K. Shaw, On the spectrum of a singular Hamiltonian system, submitted.
[11]
V. I. Kogan and F. S. Rofe-Beketov, On square-integrable solutions of symmetric systems of differential equations of arbitrary order, Proc. Royal Soc. Edin. 74A (1974), 5-39.
[12]
B. M. Levitan and 1. S. Sargsjan, "Introduction to spectral theory: selfadjoint ordinary differential operators," English translation in Translation of Mathematical Monographs 39 (1975) (Amer. Math. Soc., Rhode Island, 1975).
[13]
R. M. Kauffman, T. T. Read, and A. Zettl, "The Deficiency Index Problem for Powers of Ordinary Differential Expressions," Springer-Verlag Lecture Notes in Mathematics vol. 621, Berlin, 1977.
[14]
P. W. Walker, A vector-matrix formulation for formally symmetric ordinary differential equations with applications to solutions of integrable square, J. London Math. Soc. (2),9 (1974), 151-159.
This page intentionally left blank
This Page Intentionally Left Blank
Spectral Theory of Differential Operators I.W. Knowles and R. T. Lewis (eds.) © North·Holland Publishing Company, 1981
TWO PARAMETRIC EIGENVALUE PROBLEMS OF DIFFERENTIAL EQUATIONS C. Hunter Department of Mathematics and Computer Science Florida State University Tallahassee, Florida U.S.A.
Mathieu's equation and the angular spheroidal wave equation both lead to problems in which the eigenvalues a depend on a parameter q. The eigenvalues are analytic functions of q with simple branch points in the complex q-plane. Analytical, though approximate, relations between a and q are derived using asymptotic methods of WKBJ type. These relations appear to be valid uniformly throughout the complex q-p1ane. They predict the locations of the branch points, and reproduce a known result concer~ing the instability intervals of Mathieu's equation. MATHIEU'S EQUATION Mathieu's equation is d2y/de 2 + (a - 2q cos 2e)y = 0, (1) and its eigenfunctions are the solutions that are periodic of period 2n. For varying q, the operator of Mathieu's equation belongs to a selfadjoint holomorphic family of type (A) as defined by Kato (1966). When q = 0, the eigenvalues and eigenfunctions are simply a = n2 , y = cos or sin ne, n = nonnegative integer. For q f 0 , they form four separate classes because the cos 2e term in Mathieu's equation causes the Fourier series of the eigenfunctions to be composed of either cosine terms only or of sine terms only, and with arguments that are either even multiples of e only or odd multiples of e only. The four classes of eigenfunctions are therefore designated even cosine, odd cosine, even sine, and odd sine. The even cosine eigenfunctions, for instance, have the form
I
y(e,q) = A(2n)(q) cos k=O 2k
2ke, for a = a 2n (q) with a (0) = 4n 2 . 2n
(2 )
We shall follow the custom of using the symbols ak(q) and bk(q) for the eigenvalues of cosine· and sine eigenfunctions respectively, and identify eigenvalues by the square roots of their values at q = O. The eigenvalues are distinct when q is real, except when q = 0, and their behavior has been well studied (e.g. Meixner and Schafke 1954). SPHEROIDAL WAVE EQUATION The angular prolate spheroidal wave equation is 2 (d/drd [(1-n 2) dS/d n] + [" - c n2 - m2/(1_n 2 )] S
233
0, -1 " n ,,1.
(3)
234
CHRISTOPHER HUNTER
Here A is the eigenvalue and c 2 and m are two parameters. However, m arises as an angular wave number in applications, and is usually required to be a fixed non-negative integer. The boundary conditions on S are that it be finite at n = + 1. The transformation 2
_k
S (s i n e) 2 y, n = cos 8, C = 4q, A a + 2q - '" converts the spheroidal wave equation to d2Y/de 2 + [a - 2q cos 28 - (m 2 - '4) / sin 2eJ y = 0, 0,; 8 <: 11 . (4) This equation reduces to Mathieu's equation when m = ~, but because the boundary conditions on S transform to y(O) = y(11) = 0, only the sine eigenfunctions of Mathieu's equation are related to eigenfunctions of the spheroidal wave equation. . BRANCH POINTS OF EIGENVALUES For both equations, the eigenvalues a are branches of analytic functions of q that can have singularities of algebraic type only. Now only simple branch points, at which dq/da = 0 and a specific pair of eigenvalues become equal, have been found. The nature of the branch points is illustrated in the simplest non-trival truncation of the infinite matrix equation for the Fourier coefficients of the even cosine eigenfunctions of Mathieu's equation: a -2q 0
-q a-4 -q
0
0
0
0
0
-q a-16 -q
0
0
-q a-36
O.
0
(5)
-q
A 2 x 2 truncation gives the surprisingly accurate approximate relation 2 (a - 2) 2 = 4 + 2q, a = 2 ~ (4 + 2q 2 )k2 , (6) which predicts that a and a2 , which are 0 and 4 respectively at q = 0, O become equal to 2 when q2 = -2. Actually, aO and a2 become equal to 2.0887 when q2=-2.l573.
The pattern of branch points of the eigenvalues of Mathieu's equation in the upper half q-plane, found numerically by Blanch and Clemm (1969), is displayed in Figure 1. (The lower half q-plane contains the mirror image of this pattern and is ignored in our analysis). Pairs of eigenvalues of the same class only become equal, and equalities occur between eigenvalues with adjacent subscripts only. Some branch points of eigenvalues of the spheroidal wave equation, computed by Hunter and Guerrieri (1982), are shown in Figure 2. The m = ~ branch points in this figure are b-eigenvalue branch points of Figure 1. ASYMPTOTIC ANALYSIS OF MATHIEU'S EQUATION:
THE IMAGINARY
q-AXIS
The simplest instances of branch points are those of the even eigenvalues of Mathieu's equation that lie on the imaginary q-axis. These eigenvalues are real and positive on the segments of this axis between the origin and their branch points. To analyze them, set
::!35
7WO PARAMJ:TRTC WGENl ',lLLT1; PROBLEMS
I
I
1m q
"bs7
as,1
,
20r
a s,7
bS,7
a4,6 b4,6
b4,&
10
a
"b3,5
a3,S 4,6
b3,s
a
b2,4
a2,4
b1,3"
"a
a
'02 ,
I
"
3,S
a2,4
1,3
0
-10
a4,6
I
I
10
Req
Figure 1 Branch points of the eigenvalues of Mathieu's equation in the upper half q-plane. A point labelled a l ,3 ' for instance, is one at which a l = a 3 .
20~--~~-----'--------'\-'------~\--~
1m q
+
II.~., •
...
\
\
~
II.
+
~
\ II.
-10
•\
o
\
•
\
\
-20
•
•
•
•
Re q
10
Figure 2 The 6 branch points closest to q = 0 of the eigenvalues of the spheroidal wave equation (4) plotted for 5 different values of m : _ m = 0, • m = ~, II. m = 1, +m=2,.,m=3.
236
CHRISTOPHER HUNTER
q
where both
2 = 1. S h
(7)
hand S are real and positive, rewrite Mathieu's equation as d2y/de 2 + h2 (1 - 2i S cos 2e)y = 0,
and treat h as a large parameter. y - (l - 2i S cos 2e)-1;, cos (hw),
(8)
The WKBJ solution w(e) =
f:
(1 - 2i 8 cos 2q,)1;-; dq"
(9)
is the one that satisfies the boundary condition y' (0) = 0 for an even cosine eigenfunction. Because of symmetries possessed by Mathieu's equation, attention can be restricted to the interval 0 ~ e ~ 1;-;rr. Even cosine eigenfunctions must satisfy a second boundary condition y' (~rr) = O. If this boundary condition is applied to (9), the eigenvalue relation sin [2h G( 8)J = 0, (10) is obtained, where G( S) is the real function ~7r
G(S) =1;-;
fo
(1-2i8
k
cos 2q,) 2
dq, .
(11 ) 1::
k:
Relation (10) is exact when B = q = 0 and h = (a 2n ) 2 or (b 2n ) 2 = 2n. It breaks down however as S increases, and is unable to describe the branch points. The breakdown of relation (10) is caused by the turning pOint e = eO = 1;,rr + ~i sinh- l (1/2 S),
(12 )
in the complex e-plane at which (1 - 2i B cos 2e) vanishes and w(e) has a branch point. Figure 3 shows the configuration of Stokes and anti-Stokes lines in the e-plane when S = 0.4. (Our terminology is that Stokes lines are ones along which WKBJ solutions vary purely exponentially, while anti-Stokes lines are ones along which the variations are purely oscillatory.) The most straightforward method of correcting the eigenvalue relation (10) is via the use of Heading's (1977) global phase-integral methods in which, following Stokes' (1857) original idea, one adopts the convention that WKBJ solutions change discontinuously across Stokes lines. When Heading's rules are applied to the eigenfunction (9), one obtains the formula
J,
(13) y - (1 - 2i i3 cos 2e)-1;, [ cos(hw) + 1;-;i exp [ih(2G( S) - w)-2hH( S )J for the form of this eigenfunction to the right of the Stokes line through e=1;,rr. The new function H( B ) tha t has been i nt roduced here is H( S) = 1m [
f: o
(1 - 2i B cos
2,d
2
d<j>] .
(14)
When the boundary condition y'(1;-;rr) = 0 is applied to the solution (13), one obtains the corrected, though still approximate, eigenvalue relation of sin [2h G( B)J = l:i exp [-2h H( B)J . (15a) A similar analysis of even sine eigenfunctions yields the eigenvalue relation sin [2h G( B )J = -1;-; exp [-2h H( B)J . (15b)
237
TWO PARAMlcTRIC EIGliNVALUE PROBLEMS
~~
~~.
TT-
: I
e
0
I
Fi gure 3 Stokes lines (full) and anti-Stokes lines (dashed) in the e plane for the case B
=
0.4.
0.8~----~--~----------r----------.--------~
13
I3 cn t I------\--+----\--+----~--I----~--+-I 0.4
o
2
4
6
h
8
Figure 4 Real eigenvalues for pure imaginary q plotted in the h - B plane. The branch points, at which iqi is maximum, are ringed. The full curves are exact, while the dashed curves are plotted from the asymptotic formulae (15).
238
CHRISTOPHER HUNTER
As Figure 4 shows, these approximate relations are accurate and allow adjacent eigenvalues to become equal. The branch points plotted in Figure 1 are points 2 at which Iql = B h is the greatest, and are ringed in Figure 4. The ringed points tend, with increasing h , to lie closer to the critical value S = B 't= cn 0.S811 at which H( B) = 0 but H' (S cn't) < O. The significance of this critical value of B is as follows. The turning point 8 0 lies far above the real o-axis when S is small, but descends towards this axis as S increases. When B = Bcrit ' the two downgoi ng anti -Stokes 1i nes from 8 0 pass through 8=0 and 8 = 1;;n. Now H( B) > 0 when B < Bcrit and is large when B is small. When the exp [-2h H( B )J terms in relations (lS) are insignificant, cosine and sine eigenvalues are indistinguishable in Figure 4. But exp [-2h H( B)J is 1 when S = Bcrit' while real solutions of (lSa) and (lSb) cease to be possible when it exceeds 2. Thi s happens for B only s 1i ght ly in excess of Bcri t when his large. ASYMPTOTIC ANALYSIS OF MATHIEU'S EQUATION:
THE GENERAL CASE
The analysis that is needed for general values of q in the upper half q-plane is also based on the use of WKBJ solutions and the turning point 8 , defined 0 now as the root of a = 2q cos 28 0 for which 0 < Re(8 0 ) < 1;;rr, again plays a crucial role. Although this turning point is no longer constrained to lie on the line Re(e) = 'on, the configuration of Stokes and anti-Stokes lines from it may still be such that a single Stokes line from eO intersects the real 8-axis between e = 0 and 0 = 1;;rr. Assuming this to be the case, and applying Heading's rules at the Stokes line and the relevant boundary conditions at 8 = 0 and 8 = 1;;rr , the following eigenvalue relations are obtained for the four classes of eigenfunctions of Mathieu's equation: even cosine, (16a) sin 1;; exp [-JJ (16b) exp [-JJ sin even sine, -~ -~; exp [-JJ cos odd cosine, (16c) exp [-JJ (16d) cos odd sine, ~i The quantiti es and J , which replace the 2hG and 2hH terms in (lS), are defined by the following integrals:
J
~'IT
o (a-2q
cos 2¢)
~ 2
d¢ = I(a,q),
rOo " J (a-2q cos 2¢) 2 d¢ = 1;;[l(a,q) + iJ(a,q)J.
o
(17)
(The location of the turning point and its Stokes lines depend on the eigenvalue, so that it is necessary to check our earlier assumption for self-consistency.) The general eigenvalue relations (16a-d) are effective in locating the other branch points. Further, these other branch points, like those on the imaginary q-axis, lie close to critical stages, where critical stages are defined now as occurring when: (i) (ii)
Anti-Stokes lines from the turning point
0 go to both 8 = 0 and e = 1;;n. The appropriate one of the eigenvalue relations (16) is satisfied. 8
These two requirements restrict the possible values of and J; instance, must be an integer multiple of "i for even eigenvalues.
J, for The critical
239
1'1\'0 PARAMETRIC EIGENI'ALUE PRORLl'AIS
stages must ultimately be located numerically because expressions for and J as functions of a and q involve complete elliptic integrals. However, some elementary analytical approximations for I and J , which technically are valid for small values of (q/a) only, are effective both in providing good initial approximations for the iterative determination of critical stages and, more importantly, in providing a true accounting of the pattern of branch points as shown in Figure 1. [See equations (22) below.J The branch points must also be located numerically. The critical stages provide excellent initial approximations for their iterative determination, and are indistinguishable from the branch points in Figure 1. Relations (16) remain useful as iqi ~ = , because the assumption about the turning point and the Stokes line that was made in deriving them remains valid. In fact, as iqi ~ = , 8 0 tends either to 0 and 1;'11. To see what is happening analytically, consider the even cosine relation (16a) for instance, and rewrite it in the form: (18) exp [iIJ - exp [-iIJ = i exp [-JJ The right hand side is small when (q/a) is small. All three terms of (18) are significant at the intermediate values of q at which branch points occur. As iqi ~ = however, one or other left hand side term becomes negligible. If the second of these is negligible, then exp [i(I-iJ)1 - i , 8 0 ~ ),11, and a - -2q as in a standard large-q asymptotic expansicn (Meixner and Sch~fke 1954). Similarly, when the first term of (18) becomes negligible, then exp [-i(I + iJ)J ~ -i , 8 ~ 0 , and a ~ 2q. Relations (16) therefore appear to be valid uniformly 0 throughout the upper half q-plane. This is possible because iq/ai is never large when a is large, and our basic assumption is that a is large. ESTIMATE OF THE INSTABILITY INTERVAL Although exp [-JJ is technically asymptotically negligible when (q/a) is small, yet retaining it allows the small difference between the cosine and sine eigenvalues of the same order to be estimated correctly. This difference corresponds to an instability interval when q and a are real. It can be computed, for either even or odd eigenvalues, on the basis of relations (16) using the approximati ons I = !:i1l ak2 [1 + O(q 2/a 2 )J , J = a '.2 Iw(4ia/q) -2J[1 + O(q 2/a 2 )J . (19) The difference in the exp [-JJ terms in the equations for the cosine and sine eigenvalues an and b both of which are almost n2 , gives n an - b - 4nin n 11
exp [-JJ
~ 4n [~Jn 4n2
(20)
11
This result is the large - n form of an exact result due to Levy and Keller (1963). It has also recently been obtained by Harrell (1981) in an asymptotic study of instability intervals. ASYMPTOTIC ANALYSIS:
THE SPHEROIDAL WAVE EQUATION
The asymptotic analysis of the spheroidal wave equation is also based on the approximation that a is large, and that iq/ai and m2 do not exceed 0(1). Hence, to a sufficient degree of approximation, the spheroidal wave equation reduces to Mathieu's equation except near the ends 0 and 11 of the range of e.
240
CHRISTOPHER HUNTER
Another approximation must be used in the regions near these ends. When this is done, the eigenvalue relations (16b) and (16d) generalize to sin [I - ~ mTI + \n] -~ exp [-J + iTI(\ - ~ m)] (2la) and cos [I - ~ mTI + \n] = y,i exp [-J + in(\ - Y, m)J , (2lb) respectively. Critical stages may again be located from these approximate eigenvalue relations, and the exact branch points of the eigenvalues a(q) that lie near them can be computed numerically. The elementary analytical approximations for I and J that were mentioned earlier can be used to provide rough approximations to the critical stages. When used with the eigenvalue relation (21b) for instance, they predict critical stages at \q\
=
~ (4N + l~ + m)2, arg q
=
e
n(y, -
~N-+~mm+-l~/6)
(22a)
where N is any non-negative integer and M is any even integer in the range -2N ~ M ~ 2N , and also at \q\
=
42 e
(4N + 25 + m)2 , 6
arg q
=
Y, m+- 25/6 \ ) ' n ('~ - M 4N - +m
(22b)
where M is now any odd integer in the range -2N-l ~ M $ 2N + 1 . Both formulae correctly predict the outward movement and counterclockwise rotation of the branch points with increasing m that is seen in Figure 2. CONCLUDING REMARKS Further details of the work that is described here can be found in Hunter and Guerrieri (1981, 1982). This work is heuristic and formal proofs are lacking. Yet the directions for further studies, and ways in which presently available rigoroUS results can be improved, are clearly indicated. For instance, both Meixner and Schafke and Kato show that the radii of convergence of the small-q power series for the eigenvalues an and bn of Mathieu's equation, which are determined by the locations of the branch points, exceed (n-l) for n > 2. The asymptotic analysis and the numerical results shows the radii of convergence to 2 be O(n). This work has been supported in part by the National Science Foundation under grant MCS-7728148. REFERENCES [lJ
Blanch, G. and Clemm, D.S., The double points of Mathieu's differential equation, Math. Compo 23 (1969) 97-108.
[2]
Harrell, E. M., On the effect of the boundary conditions on the eigenvalues of ordinary differential equations, Amer. J. Math., (1981) in press.
[3J
Heading, J., Global phase-integral methods, Quart. J. Mech. Appl. Math. 30 (1977) 281-302.
[4]
Hunter, C. and Guerrieri, B., The eigenvalues of Mathieu's equation and their branch points, Studies in Appl. Math. (1981) in press.
TWO PARAMFJ'R1C EiGENVALUE PROBLEMS
241
[5J
Hunter, C. and Guerrieri, B., The eigenvalues of the angular spheroidal wave equation, Studies in Appl. Math. (1982) in press.
f6J
Kato, T., Perturbation Theory for Linear Operators (Springer, Berlin, 1966)
[7]
Levy, D. M. and Keller, J. B., Instability intervals of Hill's equation, Comm. Pure Appl. Math. 16 (1963) 469-476.
[8J
Meixner, J. and Schafke, F. W., Mathieusche Funktionen und Spharoidfunktionen (Springer, Berlin, 1954)
[9J
Stokes, G. G., On the discontinuity of arbitrary constants which appear in divergent developments, Trans. Cambridge Philos. Soc. 10 (1857) 106-128.
This page intentionally left blank
This Page Intentionally Left Blank
Spectral Theorv of Differential Operators I.W Knowles and R. T. Lewis (eds.) © North-Holland Publishing Company. 1981
SCHRODINGER OPERATORS IN THE LOW ENERGY LIMIT: SOME RECENT RESULTS IN L2(R4) Arne Jensen Department of Mathematics University of Kentucky Lexington, Kentucky 40506 USA
For a Schrodinger operator H = -6 + V in L2(R4) results on H in the low energy limit are given in the form of asymptotic expansions of (H - s)-l as s ~ O.
Consider a Schrodinger operator H = Ho + V, Ho = -6, in L2(R4) with V(x) = O( lxi-B) as Ixl ~ 00, S > 2. We give results on the low energy behavior of H in the form of asymptotic expansions of its resolvent R(s) = (H - s)-l as s ~ O. The behavior of R(s) as s ~ 0 is strongly dimensionally dependent. Results in L2(Rm) are given in [lJ for m = 3 and in [2J for m ~ 5. The results for m = 4 are given here without proofs. All the necessary techniques can be found in [1,2]. Details of the lengthy computations are given in [3]. For some related results see also [4,5,6,7J. We use the weighted Sobolev space Hm,s(R 4 ) defined for any m, s E R by Hm'S(R 4 ) = {f E S' (R4) I II (1 - 6)m/2 (1 + x2)s/2 fIIL2 < oo}. Here S' (R4) denotes the tempered distributions. We write Hm,s instead of Hm,s(R 4 ); (', .) denotes both the inner product on L2(R4) and the natural duality ms -m - s m s m' s' ms between H' and H ' . B(H', H ' ) denotes the bounded operators from H ' m to H ' ,s' , with the operator norm. Consider first H o
=
-6.
R (d o
=
(H
0
- d- l is given by
sl/2 (1) 1/2 "IT 2"IT Ix - y I H 1 (s Ix - y I ) , where H(i)(z) is the first Hankel function. Here T: k(x,y) shows that the operator T has the integral kernel k(x,y). Using the expansion for the Hankel function one obtains formally
where the operators G~ are given as follows: J
::'43
244
ARNE JENSEN
GO: (41T 2 r l /x_y/-2; Gl " o. o 0 2 (41Tr (-4)1-j((j-l)l'jlr l , j .::.1, and let 'l'(j) denote the digamma
Define c. J function.
c.{'I'(j) + 1jI(j+1) + irr}/x_y/2 j -2 - 2c. In(~/x-y/)·/x_y/2j-2,
G~ J
J
J
_c./x_y/2 j -2.
Gl J
J
The precise result on R (t;) is: o LEMMA 1. We have in the operator norm on B(H- l ,s,H 1 ,-s') the expansion R
o as
1; ...,.
O.
~
~. t;J ( 1n
L L
(d
j"O k=O
Here ]l(0)
k
d kG. + 0 (I;
~
(1 n d]l
() 9,
J
J
0, ]l(9,) " 1 for 1 .::. 1, and s, s' must satisfy 1
0
if 9, "0: 0
2
if
S,
£ >
s'
>
1/2, s + s' > 5/2;
1: s, s'
>
2L
Assumption of V. V is multiplication by a real-valued function V(x) such.that V defines a compact operator from H1,0 to H- l ,8 for some B > 2. H " Ho + V is the iU~drati:l f~:m sum. compact operator from H' to H ' S
Note that for every s E R, V is a
An expansion for R(t;) (H - t;)-l can be obtained from Lemma 1 and R(d = (1 + Ro(dVrl Ro(d, if we can obtain an expansion for the operator (1 + Ro(t;)vt1. We have 1 + Ro(dV
If 1 + GOV is invertible in B(H l ,-s,H l ,-s), 0 < S < S - 2, 0 is said to be a o regular point for H. In this case 1 + Ro(t;)V is invertible for small t;, and the inverse can be found using the Neumann series. This leads to an expansion for (1 + R (t;)V)-l with explicitly given coefficients. o THEOREM 1. Let 0 be a regular point for H. Then 9, j. k ( R ( ln ~r)£) as I; ...,. 0, R(d" L L t;J (1 n d Bjk + 0,1; j=O k"O in B(H-l,s,Hl.- s ') for£ .::.1, S > 4£, and s, s' > 2£. The first few coefficients Let X " (1 + G~V)-l.
are explicitly given as follows.
B~ = XG~X*, B~ = -XG~VXG~X*, B~ B~ = XG~X* - XG~VXG~X*.
=
XG1x*-
XG~VXG~X*
-
B~
=
XG~, B~
XG~X*,
XG~VXG~X*,
If 1 + G~V is not invertible, some further results are needed before we can find an expansion for (1 + Ro(z;)V)-l. For 0 < s < 8 - 2 let
245
SCHROEDINGER OPIiR/1TORS IN THE LOW ENERGY LIMIT
M = {f E H1 , - s
-
I
(1 + GOV) f = O}. 0
M is independent of s in the given interval. Since G~V is compact, M = {OJ generically. Precisely, consider H(x) Ho + xV, x real. Then !i(x) = {OJ except for a discrete set of values of x. LEMMA 2. For 0 < s < 2 we ha ve M = {g E H1 , - s I (H o + V) gO}, LEMMA 3. Let u EM. Then u E L2(R4) if and only if
Algebraic null space (1 + G~V) = geometric null space (1 + G~V).
Let Po denote the eigenprojection for eigenvalue zero for H. P = 0 if zero 2 0 is not an eigenvalue for H. Lemma 2 shows PoL C ~ and Lemma 3 shows dim(M\P L2) -< 1. Any u E -M\P 0 L2 is called a zero resonance wave function. These 0 results are similar to results in L2 (R 3 ), see [1]. Lemma 4 allows us to decom1 -s pose the space H' using the natural projection onto the algebraic null space for 1 + GOV. We give the leading terms in the expansions below. The technique o used allows one to compute the coefficients in the expansions to any order. Note also that we obtain finite expansions to any order by using the function (a - ln z;r l , see below.
o is
said to be an exceptional point of the first kind for H, if dim(!i) and we can find ~ E !i with 6. Then we have in B(H-l,s ,H l ,-s') R(z;) = - z; - 1 (a - 1n z;) - 1 < . ,~)~ + 0 (1 ) a is given by (y is Euler's constant) a = ni + 1 - 2y - (4nr2
If
Assume 6 as
1; ->-
>
THEOREM 3. Let 0 be an exceptional point of the second kind for H. and s, s' > 6. Then we have in B(H- l ,s,H l ,-s') R(r;) = _I;-lp + ln 1; P VG 1Vp + 0(1) 2 0 o 0 If dim(!i) ~ 2 and we can find ~ E ~ with
1 and Assume S as
I; -+
O.
a is given in Theorem 2.
12
>
O.
an exceptoo compli-
THEOREM 4. Let 0 be an exceptional point of the third kind for H. Assume S and 5, 5' > 6. We then have in B(H- 1 ,5, H1 , - s' ) R( 1;) = - I; - 1Po - I; - 1 (a - 1n I; r 1 < . , ~)~ + lnl;poVG1Vpo + 0(1) -+
O.
In(~lx-yl)·(V~)(x)(V~)(y)dxdy.
o is said to be an exceptional point of the second kind, if dim(M) !i = PoL2, i.e., ~ consists of eigenvectors for eigenvalue O.
as z;
12
>
>
12
246
.1 RNE JENSEN
REMARKS. Expansions to any order with explicitly given coefficients can be found using the techniques from [1,2]. GenerallY,expansions to higher orders require larger Band s, s'. The above results can be used to derive results on the time-decay of the wave functions, and asymptotic expansions for the scattering matrix in the low energy 1imit. REFERENCES [1]
Jensen, A. and Kato, T., Spectral properties of Schrodinger operators and time-decay of the wave functions. Duke Math. J. 46 (1979) 583-611.
[2J
Jensen, A., Spectral properties o~ Schrodinger operators and time-decay of the wave functions. Results in L (R m), m ~ 5. Duke Math. J. 47(1980),57-80.
[3]
Jensen, A., Spectral properties o~ S~hrodinger operators and time-decay of the wave functions. Results in L (R). Preprint, University of Kentucky, 1980.
[4]
Murata, M., Scattering solutions decay at least logarithmically. Japan Acad. Ser. A Math. Sci. 54 (1978) 42-45.
[5]
Murata, M., Rate of decay of local energy and spectral properties of elliptic operators. Japan. J. Math. 6 (1980) 77-127.
[6]
Rauch, J., Local decay of scattering solutions to Schrodinger's equation. Commun. Math. Phys. 61 (1978) 149-168.
[7J
Vainberg, B. R., On exterior ell iptic problems polynomially depending on a spectral parameter, and the asymptotic behavior for large time of solutions of non- s ta tiona ry problems. Ma th. USSR Sborn i k 21 (1973) 221- 239.
Proc.
Spectral Theorv of Differential Operators I.W. Knowles and R. T. Lewis (eds.) © North·Holland Publishing Company. 1981
LONG-TIME BEHAVIOR OF A NUCLEAR REACTOR* Hans G. Kaper Applied Mathematics Division Argonne National Laboratory Argonne, IL 60439
A fundamental problem of reactor physics is the determination of the long-time behavior of the neutron population in a nuclear reactor. In particular, one is interested in the question whether the total neutron density has a purely exponential behavior as t ? "". We formulate this problem as an abstract Cauchy problem, show that the solution is given by a semigroup, and investigate the asymptotic behavior of the semigroup. 1.
INTRODUCTION
A fundamental problem of reactor physics is the determination of the asymptotic behavior of a nuclear reactor for large times. Inside a reactor (a hi gh 1y heterogeneous compos ite structure of many different materi a1s) neutrons The neutrons move about freely (i .e., are generated by fission processes. rectilinearly and with constant velocity) until they interact with a nucleus of the reactor material; in the course of an interaction a neutron may disappear entirely (absorption), it may change its velocity (scattering), or it may trigger a fission process, as a result of which one or more new neutrons appear. The relevant space and time scales are such that interactions can be viewed as localized and instantaneous events. The equation that describes the rate of change of the neutron density inside the reactor is a linear transport equation; the dependent variable is the neutron velocity distribution function (f). If n denotes the reactor domai n (a bounded open convex subset of It 3), and Sis the neutron velocity range (a ball or spherical shell centered at the origin in ~3), then f(x,~,t)dxd~ represents the (expected) number of neutrons in a volume element dx centered at a point x ( n whose velocities lie in a velocity element d~ centered at the velocity t; E S at time t. The linear transport equation is a balance equation for f over the element dxdt; about (x,~), (1.1 )
if = -
~x • t;f(x,t;,t) - h(x,t;)f(x,t;,t)
+
J k(x,t;+t;')f(x,t;',t)dt;', S
x
E
(1,
s
E
S,
t
>0
The first term on the right is the (spatial) divergence of the neutron flux, which represents the effect of the free streaming; the second term represents the loss due to interactions at x, h(x,~)dt; being the collision frequency for neutrons with the velocity in the range dt; about t; at the point x; the third term represents the gain due to interactions at x, k(x,~+~' )dt; being the (expected) number of neutrons emerging with a velocity in the range d~ about t; after an
*Joint work with C. G. Lekkerkerker (U. of Amsterdam, Neth.) and J. Hejtmanek (U. of Vienna, Austria). This work was supported by the Applied Mathematical Sciences Research Program (KC-04-02) of the Office of Energy Research of the U.S. Department of Energy under Contract W-31-109-Eng-38. 247
HANS C, KAFER
248
interaction of a neutron with the velocity ~' with a nucleus of the reactor material at x. With Eq. 1.1 are prescribed an initial condition,
(1.2 )
(x, 1;)
E
rlxS ,
and a boundary condition on arl. The boundary condition expresses the fact that no neutrons enter the reactor from outside ("zero incoming flux"); it may be formulated as (1.3) where Sx
f(x,l;,t)
0
= {I; E S: x
t
+ t[,
E
Q for some t
> oj,
X E
>
°
an.
The quantity of interest is the total neutron density inside the reactor, i.e., the integral ~ f(x,l;,t)dxdl;; in particular, its asymptotic behavior as t + 00. For practical purposes one wants to know under which conditions on the functions hand k the integral behaves like a pure exponential as t + "'. We might add that, for many reactor materials, the functions hand k vary rapidly with the neutron velocity: they may display resonances, etcetera. As we shall see, a satisfactory solution to this problem has not yet been given. Partial answers are available, 'and new results from the theory of strongly continuous semi groups of positive operators in Banach lattices are being applied.
$
In the next section we give the functional formulation of the reactor problem as an abstract Cauchy problem. In Section 3 we show that this abstract Cauchy problem is solved by a strongly continuous semigroup of positive operators. In the final Section 4 we discuss some results about the asymptotic behavior of the semigroup. Details of the proofs, as well as related results, can be found in our forthcoming monograph [1, Chapter 12J. 2.
FUNCTIONAL FORMULATION
Let Q be a bounded, open, convex subset of 11 3 , and let S be a ball of finite radius centered at the origin in 11 3 In this section we shall show that the initial-boundary value problem 1.1-3 leads to an abstract initial value problem for the function f: [0,00) + Ll(rlxS). (The choice of an Ll-space is a natural one in the present context, as f is nonnegative and its L1-norm gives the total number of neutrons inside the reactor.) We begin with the definition of the collisionless transport operator (-T), which corresponds to the first term in the right member of Eq. 1.1. Two technical difficulties arise: one because the expression (a/ax)"l;f is singular at 1;=0, the other because the boundary condition 1.3 involves only part of the range of the variable 1;. Let C O(QxS) be the space of all functions f that satisfy the conditions (i) supp fCQxSaB for some S2, a> 0, where Sas = {I;EIl3: a~ 11;1 ~ sl; and (i i) f admits a {B,E)-extension to QExS for some E > 0; here, QE is a E-neighborhood of Q, and a (B,c)-extension is a function fEE COO(QExS) whose restriction to rlxS coincides with f and which vanishes on each incoming ray up to a point inside Q (i.e., for each (x,l;) E QxS, let T = T(x,l;) denote the unique nonnegative number such that x-TI; E aQ; then there exists a 1'1 E (O,T) such that f,O<x-sl;,i;) = for all s > 11.) Let TO be defined in CB' O(QxS) by the express i on '
s
°
(2.1)
(x,l;)
E
QxS,
f
e
C~,o{QxS) .
LONG TIMF BEHA V[()UR OF A NUCLEAR REACTOR
249
s
Then \I+TO(AE[) is a bijective map of C O(QxS) onto itself. If Re\ ) 0, then (\I+T O)-l can be extended by continuity to a bounded linear operator RA in L1 (QxS), where (2.2)
RAg(X,~) =
T
e-ASg(x_s~,~)ds ,
f o
for almost all (x,~) E QxS. This operator R\ is injective; its inverse is the closure of AI+TO' so if we define T by (2.3)
T = R\
1
AI ,
-
then T is uniquely defined and T is the closure of TO' The second and third term in the right member of Eq. 1.1 give rise to bounded linear operators in Ll(QxS), provided h E L""(QxS) and hp E L""(QxS), where hp(x,~I) = 1 k(x,~<-~')d~ for (x,~') f QxS. We shall assume that these conditions are met, ana define the operators Al and A2 in Ll(~xS) by the expressions (Z.4 )
Al f(x,~)
(Z. 5)
A f( Z
for any f
E
h(x,~)f(x,n
f
x, ~)
k(x ,~+C )f( x,~ I )d~
I
(x,!;)
E
QxS
(x, 1;)
E
QxS
S
Ll(~xS).
Then IIAIIl = IIhll"" and IIAZII = IIhpll"".
The initial-boundary value problem 1.1-3 thus gives rise to the following abstract Cauchy problem in Ll(~xS): (Z .6)
3.
f'(t) = (-T-A +A )f(t) , 1 2
t > 0;
f(O)
SOLUTION OF THE ABSTRACT CAUCHY PROBLEM
We consider the transport operator - T-AI +A Z as a pert urb at i on of the stre aming operator -(T+A 1 ) by the bounded operator AZ ' The spectrum a(-(T+A 1 )) is determined by the behavior of h(x,~) for small values of 11;1. Let the nonnegative constant \* be defined by (3.1)
A* = inf{ lim
I~I"O
h{x,I;):
x E ~}
Assume that (A) There exists a positive constant c such that and a 11 I; C S wi th I; of O.
h(x,~)
> \*-cll;l
for all x
E
n
Then the right half-plane {A E [: ReA> -\*} belongs to the resolvent set p(-(T+A1)); moreover, the resolvent QA = (AI+T+A1)-1 sat.isfies the estimate IIQ~II ~ M(ReHA*)-n for n=1,2, ... , with M = exp(c·diam~). It follows from the Hille-Yosida theorem [Z, Section IX.1] that -(T+A 1 ) is the infinitesimal generator of a strongly continuous semigroup WI = [WI (t): t 2. 0] in Ll(~xS). The expression for WI (t) is readi 1y found,
250
HANS C. KAPliR
t exp(-J h(x-s~,~)ds)g(x-s~,~)
(3.1 )
(x,~)
o
E
IlxS ,
for any g E Ll(llxS). The semigroup consists of positive operators. As the underlying space is an Ll-space, the type of the semigroup coincides with the spectral bound of the generator [3, Section 3.3J. The latter is at most equal to -A*; it is exactly equal to -A* if we assume, in addition, that (B) For each E > 0 there exists a ball BO and a constant n > 0 such that h(x,~)
~ p} wholly contained in 11 A*+E for all x E: BO and ~ E S with
! Ix-xOI <
I ~ I < n. In fact, if (A) and (B) hold, ReA.s. -A*}.
then a(-(T+A 1 )) fills the entire half-space {A
E
[:
We now add the bounded perturbat i on A2 to - (T+A 1 ). Accord i ng to the theorem of Hille and Phillips [Z, Section IX.2.1J, the resulting operator -(T+A 1 )+A Z is the infinitesimal generator of a strongly continuous semi group W = [W(t): t > OJ in Ll(llxS). This semigroup provides the solution of the abstract Cauchy prob~m. THEOREM 1. given by (3.2)
~
fO ( domT, then the solution of
f(t) = W(t)f O '
t
>
~ ~ ~
uniquely determined
~
°.
The semigroup W cannot be determined explicitly. However, W can be found from Duhamel's integral equation t (3.3) W(t) = WI (t) + J WI (t-s)A 2 W( s)ds , t.2. 0 ,
°
by iteration; the result is the following Dyson-Phillips expansion:
Y win)(t), t > 0 , n=O where wfO)(t) = W1 (t), wfn)(t) W1 (t) + /Wl(t-S)A2Wfn-1)(s)dS for n=I,Z, .... ·34 . The ser1es . converges 1n the operator norm topology, see [2, Section IX.Z.1J. Because AZ is a positive operator, the semigroup W consists again of positive operators. (3.4)
W(t) =
°
4.
ASYMPTOTIC BEHAVIOR
The type of the semi group W coincides with transport operator. We denote the latter by AO'
the
spectral
bound
of the
It follows from the general theory of strongly continuous semigroups of positive operators that AO E a(-(T+Al)+A Z)' see [3, Section 3.4J. The perturbation AZ is a partial identity in Ll(llxS), so it is certainly not compact. However, the operator AZWl(t)AZ is an integral operator in Ll(llxS),
251
LONG TIME lJHI.I LFJl'R ()FA NL'CLL/IR REACTOR
(4. Z)
! !Ht(x,t:,x',t:')f(x',I;')dx'dl;',
"s where =
I
~
k(x,t;
t x
+
x-x'
--t--)k(x',
t exp (J -
o
h ( x-s
x-x'
--t-- + 1;'
XX' XX')) T' --T- ds
) t
,
>0
.
The representation 4.Z enables us to use compactness arguments. ~
THEOREM Z. pact for ~
n
for some positive integer n, the product i~1 (WI (t i )A Z) is
com-
(tl,t Z "" ,tn) ~ ~ function continuous in the uniform operator topology, then {A C [: ReA = -A*} C a(-(T+A 1 )+A Z)' ~ AO ~ -A*; if AO > -A*, then a(-(T+A 1 )+A Z) contains finitely many points Ak (k=O, ... ,m) l!!. each right half-plane ReA> -A* + s (s > 0), each .2..!. these points ~ ~ eigenvalue of -(T+A1)+A Z with finite algebraic multiplicity, and n-tuples
of
positive
n [i~1 (WI(ti)A Z ): (t1,t Z,···,t n ) c
(4.3)
m
W(t)
I
1121
numbers
~
\ t tOk
e
e
Pk + Zn (t) (I -P) ,
k=O where IIZn(t)II = o(exp(-A*+s)t) ~ t -> 00; here Pk and Ok ~ the projection and nilpotent operator associated with Ak' ~ P = PO+" .+P k . The representation 4.3 can be sharpened if one can show that the semi group W is irreducible. In the present context, W is irreducible if there exists a to > 0 such that W(t) is positivity improving for each t ~ to' Indeed, if W is irreducible, then AO is a simple eigenvalue, the projection Po is positivity improving, and there exists as> 0 such that the real part of any other point of a(-(T+A1)+A Z) is less than AO-s. Thus, Aot (4.4) W(t) = e Po + Z(t)(I-P ) , O where Z = [Z(t): t ~ OJ is a semigroup in (I-P O)L 1 (>2 x S). Although the spectral bound of the generator of Z is strictly less than AO' one can only conclude that the type of the semi group Z is less than or equal to AO' as Z does not necessarily consist of positive operators. REFERENCES [lJ Kaper, H. G., Lekkerkerker, C. G., and Hejtmanek, J., Spectral Methods in Linear Transport Theory (Birkhauser Verlag, Basel, to appear) [ZJ Kato, T., Perturbation Theory for Linear Operators York, 1966) [3J Derndinger, R., (1980), Z81-Z93.
Ueber
das
Spektrum
(Springer Verlag,
positiver Generatoren,
Math.
Z.
New 172
This page intentionally left blank
This Page Intentionally Left Blank
Spectral Theory of Differential Operators I.W. Knowles and R. T. Lewis (eds.) © North·Holland Publishing Company. 1981
REMARKS ON THE SELFADJOINTNESS AND RELATED PROBLEMS FOR DIFFERENTIAL OPERATORS Tosio Kato* Department of Mathematics University of California, Berkeley
This will be a partial survey, with some new results included, of recent results on (essential) selfadjointness problems and their generalizations for linear differential operators. The main topics will be: the (essential) selfadjointness of second-order elliptic operators with oscillating potentials; the (essential) selfadjointness of higher-order elliptic operators; characterization of the domain for nonnegative potentials; the m-accretivity and mdispersiveness of degenerate-elliptic operators of second order in LP (Rm). 1.
Introduction. This is a partial, rather incomplete survey of recent results on the problems of (essential) selfadjointness for linear differential operators and their generalizations (such as quasi-m-accretivity). It consists of a review of more or less random samples of those recent results which are known to me, together with various comments and remarks, including some new results of our own. I can only apologize for any possible omission of other important results. In section 2 I review the (essential) selfadjointness in L2(Rm) of secondorder elliptic operators with variable coefficients. The emphasis is on the global oscillatory behavior of the potential, rather than its local singularities. Section 3 discusses a recent definitive result, due to Leinfelder and Simader, on Schrodinger operators with singular vector potentials. In section 4, I consider second-order, degenerate elliptic operators with real coefficients, which need not be formally selfadjoint. The main problem is the essential quasi-maccretivity of the minimal operator and the quasi-m-accretivity of the maximal operator, in the real Banach space LP(R m), 1 < p < =. In section 5, I discuss the domain of the selfadjoint operators considered in section 2. The main question is whether or not the domain is the intersection of the domains of the "kinetic energy" part and the "potential energy" part of the operator. In the last section, I shall introduce some results on the essential selfadjointness of general higher-order, strongly elliptic operators on Rm, including the domain problem just mentioned. In Appendices I give a proof of a theorem stated in section 2, together with other technical remarks. 253
254
2.
TOSIO KATO
Second-order ell iptic operators. 2.1. Second-order operators of the form m
T
(2.1)
I
j,k=l
o.a·k(x)Ok + q(x), J J
0j=dj-ibj(x), dj=d/dx j , j=l,···,m, have been studied extensively. We want to discuss some of the recent results.
In this section we make these standing assumptions: a = a kj real-valued, (2.2) a EO Lip(R m), jk jk where Lip denotes the set of locally Lipschitzian functions. (2.3)
(2.4)
The matrix (a.k(x)) is positive-definite J m b· EO Lip(R ), real-valued.
J 1 m m (2.5) q = q+ - q_, 0 ~ q+ EO Lloc(R ), o ~ q_ EO Lloc(R ). We thus assume that q is locally semi bounded, to avoid technical complications. 00
Under these assumptions, T¢ makes sense for all ¢ EO COO = Coo(Rm) but need not o 0 be in L2 = L2 (R m). If q EO L~oc' then H EO L2 and we define Tmin to be the operator in L2 given by T . ¢ = T¢ with domain O(T . ) = Coo. mln
mln
0
We define T as the restriction of T with O(T ) as the set of all u EO L2 2 max max with Tu EO L. Here Tu is taken in the distri bution sense. To make sense out of 1 the term qu as a distribution, we assume that u EO O(Tmax) implies qu EO Lloc ' T in this general setting was considered in Kato [15J in the special case max 2 o'k' b. = 0 and with q = O(lxl ) (with mild local singularities). It was a jk J J shown that Tmax is selfadjoint in L2. Since (2.6) Tmax = T~in if q EO L~oc' 2 this proves also that Tmin is essentially selfadjoint if q+ EO Lloc ' These results have been generalized by many authors, including Eastham-EvansMcLeod [7], Frehse [llJ, Oevinatz [3J, Evans [8J, Knowles [20,21J, Kalf [14J, with more and more emphasis on oscillatory potentials q. All these papers contain a local characterization of O(T ), which says that -max 1/2 2 ) implies d.u, O.U, q+ u EO Ll ' (2.7) u EO O(T max J J oc and this is essential in the proof of the selfadjointness of Tmax or the essential selfadjointness of Tmin under various additional assumptions on the global behavior of q. A particularly strong and useful result is given by Knowles [22]. There is a densely-defined operator To in L2 such that (2.8) T C T* = T . o 0 max To is called the minimal operator by Knowles, but we shall reserve Tmin in the original sense. What is important is that Tmax is the adjoint of a certain symmetric operator. This is very effective in applications, since it reduces the
SELFADJOINTIVESS PROIH.t:JIlS H)R DIFFFRLN1T1L OPI]RATORS
255
proof of the selfadjointness of Tmax to the proof that (2.9)
(Tmax
+ i)u+
= 0
implies
u+ = O.
2.2. There are some variations among these authors in the continuity assumptions on the a. k and b .. For example, Eastham-Evans-McLeod assume that 1+ J J a. E C a rather than Lip, and are followed by Evans, while Frehse assumes only k L~P. I presume that Cl +a was technically required in connection with the local singularities of q. I would conjecture that Lip is sufficient even in the presence of such local singularities, though this would require a careful study. 2.3. Among these works listed above, it seems to me that the most general sufficient condition for the essential selfadjointness of Tmin is contained in Evans [8] (although this paper has the main purpose of considering the powers Tk ). I would rather not reproduce his condition here, which is not very simple, even in the special case (2.5) I am assuming. It will only be noted that it consists of the restriction of the growth rate of the a jk , expressed in terms of an upper bound p+(r) of the largest eigenvalues of (ajk(x)) for Ixl = r, to be correlated with the growth rate of q+ in a complicated way. The condition is general enough to allow a variety of oscillatory behaviors of q. It should be noted that the assumptions of Evans imply (2.10) Joo p(rf 1/2 dr=oo, 1
+
(see Appendix 3)
although this is not explicitly mentioned in the paper. It is somewhat disturbing, in view of the otherwise very general nature of his condition. Looking into other papers, I found that very few authors gave sufficient conditions that do not imply (2.10). Frehse [11] is one of the few, and his condition regarding the growth rate of p+(r) is very mild. But he had to correlate it with P_ (r), a lower bound of the smallest eigenvalues of (ajk(x)) for Ixl = r, even when q_ = O. I am rather reluctant to introduce p_(r) into the assumptions when q has no local singularities. Actually Evans also uses p_{r), but only in connection with such local singularities of q. 2.4. Some comments are in order regarding (2.10). An analogous but stronger condition (in which p+(r) is replaced with a*(r), the supremum of the largest eigenvalues of ((a.k(x)) for Ixl ~ r) was implied by the assumptions used in J .. Ikebe-Kato [12], as was pointed out by Jorgens [13]. Jorgens was able to remove this defect, but not very substantially. In fact in Ikebe-Kato, one could have replaced a*(r) with the radial bound of (ajk{x)): m -2 a* (r) = sup arad{s), (2.11 ) I aJ·k{x)xJ.xkr sup rad l<s
TOSIO KATO
256
rather than (2.10). 2.5. Here I would like to present my own version of Evans-type condition, which has some advantage in avoiding the use of a radial variable like r, and which does not imply (2.13) even when the variable r is used. In this condition, several auxiliary functions satisfying eikonal-type differential inequalities are used. Such inequalities were previously used by Jorgens [13J and Cordes [2J. To state the theorem, I find it convenient to introduce the notation (2.14 )
f·a·g
=
f·a.g(x)
1 m) THEOREM I. Assume that there exist three real-valued functions U, V, WEHloc(R (local Sobolev space) and constants Ko'··· ,K6 < "', 0 < 0 < 1, 0 < n < 1, satisfying the following conditions: 2 (i) W ~ Ko and q_W2 ~ Kl . (ii) dW·a·dW ~ K2 + (1 - o)q+w2. (iii) dV·a·dV ~ K3w2 + K4q+w 4 , and V(x) ~ '" as Ixl ~ "'.
K5 + K6q+l-T] ' and U(x) ~ as Ix I ~ "'. Then Tmax is selfadjoint. If m = 1, U is redundant and condition (iv) should be disregarded.
(iv)
dU·a·dU
~
00
The proof of Theorem I will be given in Appendix 1. given here.
Several remarks will be
(a) The theorem generalizes Evans's theorem under the restriction (2.5). If W= W(r) is assumed to depend only on r = lxi, our condition (ii) becomes (x.a.x)r- 2W,2 ~ K2 + (1 - o)q+W2, which is implied by a rad (r)w,2 .::.K2 + (1-o)q+W 2. This is in turn implied by condition (i) in Evans's Theorem 1. Our condition (i) coincides with Evans's (ii); note that w may be assumed to be bounded in Evans's theorem. An exact analog of our (iii) appears in the form of a diverging interval in his (iv), in terms of the radial variable r. Only our (iv) has no exact analog in Evans, though his (v) may be remotely related to it. In fact (v) is a rather restrictive condition and is responsible for (2.10) or (2.13). When (2.13) is satisfied, one can take U = Jr a d(s)-1/2dS to satisfy our (iv). (cf. also 1 ra Jorgens [13J). (b) Our assumptions (i) to (iv) do not imply (2.13). Indeed, arad(r) can grow arbitrarily fast if, say, q_ = 0 and q+ grows sufficiently fast at infinity (see Appendix 2). The appearance in (iv) of ql-n rather than q+, on the other hand, is a weak point in the theorem, although it can be slightly relaxed to, say, q+. (log (2 + q+) 1- n.
r
(c) U and condition (iv) are not necessary if m is close to a theorem of Read [26J.
1.
For m = 1, the theorem
SELFAD]OINTNESS PROBLEMS FOR DIFrriRENTlAL OPERATORS
257
(d) As a general remark, one may say that a theorem of this type could not give a definitive result for selfadjointness. It corresponds to the classical mechanics, as the use of eikonal-type differential inequalities suggests, while selfadjointness is essentially a quantum-mechanical property. (e) In the proof of Theorem I (Appendix 1), I use integrations by parts after introducing cut-off functions. According to Kalf [14], one might use Gauss's theorem with boundary integrals. I was not successful in this attempt except for m = 1, in which case it does work better. 3.
Schrodi nger operators with vector potenti a 1s. 3. l. In this section we consider the operator (2.1) in which
(3.1)
ajk(x) = °jk' Most of the following results are expected to be true in more general cases in which (ajk(x)} is bounded and uniformly elliptic, but apparently details have not been worked out. To compensate for the strong assumption (3.1), we want to allow some singularities for the vector potentials bj . Such singular bj have been considered by Schechter [27], Simon [28,29], Kato [16J, and others. A recent paper by Leinfelder-Simader [23] seems to have finished the problem by giving a definitive resul t. The Leinfelder-Simader theorem consists of two parts. In the first part, they assume only b. E L2 (Rm) real, (3.2) o -< q E Lll oc (R m), loc J and prove that the closure of the minimal form hmin associated with (2.1) coincides with the maximal form h In other words, COO0is- a- core for h Here -max - - max hmax is given by m 2 (3.3) hmax[u] = j~l IID j Ul1 + Ilql/2uI12, with the domain D(h max ) consisting of all u E L2 such that the Dju and ql/2u are in L2. Here D.u = d.u - ib.u is taken in the distribution sense, which is possiJ J J 2 ble since d.u and b.u exist for u E L as distributions by (3.2). h. is by J J ml n definition the restriction of hmax with domain C~, which is obviously a subset of D(h max )' The question of hmax = h was raised in [16] and partially answered. min Simon [29J gave a stronger reSUlt, but Leinfelder-Simader's result is definitive with the weakest possible assumptions. Their proof is based on a systematic use of calculus with Wl,p_ functions. An important result (decisive in the second part) is that the selfadjoint operator H associated with hmax has a core containing only LOO-functions.
258
'[OSlO KA '[0
3.2. In the second part of the Leinfelder-Simader theory, the essential selfadjointness of Tmin is proved under the assumptions . b E L2 (m) 2 m). L4loc (Rm , )dlV 0 ~ q E Lloc(R loc R , Again, these are the minimal assumptions for Tmin to make sense. The proof is not simple but follows a natural course. (3.4)
bj
E
It seems to me that the following lemma is implicitly involved in the proof, though it is not directly mentioned in the paper. 1
4
Let u E Lloc n Hloc and fj E Lloc ' If LlU + I j 2 E L41 . (Here Hk = Wk ,2 is the Sobolev u E Hloc and d.u J oc It seems to me that a somewhat simpler proof of L-S using the lemma. The lemma can be proved by a bootstrap dju E Ljoc for p close to 2 and raising it to p = 4. LEMMA I.
00
fjdju
2
L1oc ' then space of L2-type.) E
theorem can be given by argument, showing that
3.3. One might ask if the selfadjointness of Tmax can be proved under the assumptions (3.2) only. The difficulty here is that the meaning of Tmax is not clear. On the other hand, it would be interesting to see if the above results can be generalized to include negative potentials -q with or without local singularities. 4.
Degenerate-elliptic operators of second order. 4.1. In this section we consider operators of the form m m T=(4.1) + I a.(x)d. + a(x), I d.a·k(x)d k J j=l J j , k=l J J
x E Rm,
in which all the coefficients are real-valued. Moreover, we assume. d2a jk , da j , and a are in Loo(R m); (4.2) (4.3) the matrix (ajk(x)) is nonnegative for each x E Rm. Note that (ajk(x)) need not be positive. In an extreme case a jk may be identically zero so that T becomes formally a first-order operator, which may itself be identically zero. Condition (4.2) is rather mild regarding the growth rate of a jk etc. at infinity. Indeed, a'k(x) may grow like 0(lxI 2 ) and a.(x) like O(lxl). (4.2) J J appears to be a natural assumption in the degenerate-elliptic problem, in view of a basic inequality due to Oleinik [24], which is an indispensable tool in its study. If aj = 0, then (4.1) is formally selfadjoint. The essential selfadjointness of Tmin in L2(Rm) in this special case was proved by Devinatz [4], in which
259
SELF.4DjOINTNESS PIWIlLJ:AIS FOR DlFFEIUiNrIAL OPERATORS
essential use is made of the theory of stochastic differential equations. A natural generalization of this result to the case a. f 0 would be that 2 J T is essentially quasi-m-accretive in L. It means that the closure Tmin , min which coincides with _ Tmax , is the (negative) generator of a quasi-contractive semigroup {exp(-tTmln . ); t -> OJ. (A semigroup {U(t); t ~ O} is quasi-contractive if ~U(t)1 ~ est for some real constant s.) m One might expect that these results are true in LP(R ) if 1 < P < A densely-defined linear operator A in LP (which is assumed to be real in this section) is quasi-accretive if (4.4) ((A + s)u,luI P- 2 u) > 0 for u E D(A) with some constant S. Here (f,g) denotes the pairing between f E LP and g E LP', p-l + p,-l = 1. A is quasi-m-accretive if in addition the range of A + A is all tA of LP for A > S. It is known that {e- } is a quasi-contractive semigroup on LP if and only if A is quasi-m-accretive. 00.
Thus one may raise the questions: Is T quasi-m-accretive in LP? min Is Tmax quasi-m-accretive in LP? It is a priori conceivable that the answer is yes or no for both questions, or yes for one question and no for the other. These questions are related to the p' same ones for the formal adjoint 5 of T. If we consider Smin and Smax in L (01) and (02) are equivalent (in the reversed order) to (01') Is Smin quasi-m-accritive in LP '?
(01) (02)
(02')
Is Smax quasi-m-accretive in LP'? Indeed, (01') is dual to (02) and (02') to (01) by the well-known relations (4. 5)
5
= T*.
T
= 5*. .
max mln' max mln It was shown by Devinatz [5J by probabilistic methods that the answers to these questions are yes if the coefficients a· k , a., and a have compact supports. I J J conjecture that the same is true in the general case of (4.2-3) so that one has T = T. for all p, but so far we have proved this only for (01) and (02') max mln with p ~ 2 and (02) and (01') with p ~ 2. (In any case all can be proved if oo one adds the condition da jk E L . ) 4.2. In addition to the quasi-accretivity of Tmin and related problems, there is another important notion attached to the operator T. One may ask whether or not -T is guasi-dispersive. According to Phillips [25J, a linear operator -A in LP is dispersive if (4.6) (Au,u~-l)~O for U E D(A), where u+ = max{u,O}. We shall say -A is quasi-dispersive if -A - S is dispersive for some constant s. Again, -A is quasi-m-dispersive if, in addition, the range of A + A is the whole space LP for A > s. (Actually Phillips defines dispersiveness in general
260
TOSIO KATO
Banach lattices.) According to a theorem of Phillips [25], a densely-defined dispersive operator -A with nonempty resolvent set is m-dissipative (i.e., A is m-accretive) and, in addition, the semi group e- tA is positivity-preserving. In the case of our operator (4.1), it is expected that -Tmax is not only quasi-m-dissipative but also quasi-m-dispersive, so that the semigroup generated is positivity-preserving. Since (4.6) is similar to the corresponding accretivity (dissipativity) condition (4.4), the same computation can be used to acquire this additional information. 5.
The domain characterization. Another problem related to (2.1) is an explicit characterization of the domain of T For example, consider the Schrodinger operator max (5.1) T = -t, + q(x). Given a q such that Tmax is selfadjoint, one may ask if O(T ) = D(-t,) n D(q) = H2(Rm) n O(q). (5.2) max Results of this kind are important in many problems. In the theory of evolution equations, for example, it is important to construct an isomorphism S of a Banach space Y, continuously embedded in another Banach space X, onto X. S =T is a good choice for Y = H2 n O(q) and X= L2 if (5.2) is true. max Questions of the form (5.2) have been studied by Sohr [30,31]. A convenient theorem due to Sohr is the following. Let A, B be m-accretive operators in a -1 Hilbert space H, with A bounded. Then A + B with O(A + B) = O(A) n O(B) is m-accret i ve if -1 2 for U E O(B*), Re(B*u,A u) > - c~u~ (5.3) where c
<
1 is a constant.
Applied to (5.1), (5.3) leads to the following sufficient condition for (5.2) to be true. (For similar but stronger results see Everitt-Giertz [10].) (5.4)
q ~ 0,
for some c
<
2.
This condition is extremely mild as a growth condition for q at infinity. Indeed, it is satisfied by functions such as q(x) = exp(jxjk), exp(exp(jxjk)), etc. On the other hand, it is not convenient when applied to oscillatory potentials. Similar questions for more general operators of the form (2.1) have been considered by Evans-Zettl [9]. The corresponding problem for higher-order operators will be discussed in the next section.
SELFADjOIN'INESS PROHUiMS FOR DIFFERENTI.4L OPER.-1TORS
261
6.
Higher-order elliptic operators. Consider an operator of the form T= I (-1) laldaaas(x)d B, (6.1) Ia I, I"l-==-N where a, S range over all multi-indices such that lal -==- N, lsi < N. We assume that T is strongly elliptic in the sense that (6.2) y aa~(x)~a~s ~ ol~12N, lal=Tsl=N with a constant 6 > O. The aaS are assumed to be hermitian symmetric in a, S. If the a are sufficiently smooth and bounded, it is more or less well as 2 m known that T. is essentially selfadjoint in L (R ) with T. = T (see e.g. ml n mln max Browder [lJ). A reasonable smoothness condition for the a S appears to be (6.3) a E Clal(Rm) n L=(Rm). a as In analogy with (2.5), we want to allow some singularity for at least the zeroth order coefficient a (x) = q(x). It is not difficult to show that T . 00 mln is essentially selfadjoint if (6.4) q=q+-q, q±~O, q+EL~oc' q EL=. One can further relax condition (6.4) to some extent. It has been shown by Dung [6J that (6.5) as Ix I -+ = qJx)=O(lxl) is sufficient. Related results have been given by Keller [18,19J with different assumptions. Keller assumes the highest order term in (6.1) to be (_lI)N but admits more singularities for lower-order coefficients. In any case, the admissible growth rate for q appears to decrease with N. The rate admitted by Ke ller is (6.6) For the operator (6.1) one may ask the domain question mentioned in the previ ous section-: is (6.7) O(T ) = H2N(Rm) " O(q) max true? The answer is yes under certain additional smoothness conditions on the a S and a mild restriction on the growth rate of q, for example, d5.8) q > 0, Idaql < c ql+l a l/ 2N for lal -==- N, a which is a generalization of (5.4). Here the ca are certain constants depending on the a . Although it is difficult to estimate them, (6.8) is certainly as satisfied if (6.9) q ~ 0, for la I -==- N. See [6J for these results. We note that conditions of the form (6.9) were considered, in a cruder form, in connection with the KdV equation (see Kato [17J), where it was required to find an isomorphism S that maps a certain weighted Sobolev space over (-00,=) onto L2 (_=,=).
262
TOSIOKATO
APPENDICES Appendix 1. Proof of Theorem I. According to the remark given in (Z.g), it suffices to show that (Al) Tu = iu with u E L2 implies u = O. (The eigenvalue -i can be handled in the same way.) We note that (Al) implies (see (2.7)) 2 1/2 2 (A2) DjU E L loc ' q+ U E L loc ' djU, An immediate consequence of (A2) and conditions (i)-(iv) in the theorem is 2 2 Z 2 1 Iul dW·a·dW ~ Kzlul + (1-6)q+W lui E L loc ' 2 2 Z 4 2 1 (A3) lui dV·a·dV ~ K3W lui + K4 q+W lui E L loc ' 1 ' IUIzdU·a·DU ~ KS IU12 + K6q+l-fli u 12 E Lloc PROPOSITION Al.
One has (M) J W2Du.a.Du dx ~ Kllu11 2 , J q+w2lul2dx ~ Kllul1 2 , where I I is the L2-norm on Rm, the integrals (here and in the sequel) are taken on Rm, and K is a constant depending on the Kj and 0, fl. Proof. We use the standard techniques of integration by parts. To this end, we need a family of cut-off functions given by (AS) ¢(x) = ¢E (x) = ~(EU(X)), where E > a is a small parameter and ~ is a fixed function with the following properties.
a~
~ E C~(_oo,oo),
(A6)
o<
-
cP'
(t)
<
(t) ~ 1,
(t)l-fl
for
cp(O)
t
>
=
1,
O.
To see that such a function cj> exists, it suffices to choose an appropriate and set cj> = ~~ with sufficiently large integer k. function CPo E
c;
We note that 0 < 1> < 1 and ¢ has compact support, since U(x) Since U E Hll oc ' one has ~ E Hl with (Al) d¢ = ECP' (EU)dU, (AS) d<jl·a·d<jl < E2q, 2-211 dU·a·dU < E2<jI 2-2f1 (K 5 + K6q+1-11 ) - 2 2 ~ E (K7 + KS
->
00
as
Ixl -,
In view of the local properties of u given by (A2) and (A3), the following results based on formal integrations by parts are justified. (A9)
il¢wu\l2
=
(Tu,<jI2W2u)
J [Du·a·D(q, 2w2u) + q, 2w2qluI 2]dx
f
[q, Zw20u.a.Du + 2q,2wuDu·a·dW + 2q,W 2uDu·a.dq, + q,2W2(q+_Q_)luI 2]dx.
00.
263
)ELFADjOlN'fi'VESS PROBLJ:A1S FOR DI1'l'LRENTIAL OPlc"RA'tORS
Indeed, all terms in the integrand on the right are in Ll by (A3), because ~ has compact support, with 2IWuOu-a-dWI 2. (1-o2)w 20u-a-ou + (1-6 2r l luI 2dW-a_dW 2. (1-o2)W 20u_a_OU + (1-62rlK2IuI2 + (1+6rlq+w2IuI2
by (A3),
21q,uOu-a-d~1 2. Eq,2 0u -a -l5U + E-lluI2dq,_a_d~ 2
-
2
2
by (AS)_
2. E¢ Ou-a-Ou + Elul (K7 + Ks¢ q+) Taking the real part of (A9), we thus obtain (A10) f ¢2W 2[(o2 - E:)Ou-a-l5U + (1 - (1 + l - EKS)q+lu 12]dx ~ (K l + (1 - 62 )-lK2 + EKOK7)~u~2, where we have used (i) as well.
or
We now let € + 0 and note that 1> t 1 monotonically by (A5-6) , obtaining f W2[6 20u.a.ou + 6(1 + orlq+luI 2]dx ~ (K l + (1 - 82rlK2)lluI12, which proves Proposition Al_ With Proposition Al at hand, it is now easy to complete the proof of Theorem I. We choose another cut-off function ljJ(x) = 4) E (x) = cJl(EV(X)), (All) where cP may be (but need not be) the same function as above. Again ljJ E Hl with compact support, and we have (A12) i (u,ljJu) (Tu,ljJu) 2 = f [Ou-a-O(ljJu) + QljJlul ]dx = f [ljJOu·a·Ou + uOu·a·dljJ + QljJluI 2]dx, the formal integration by parts being justified as above_ Taking the imaginary part of (A12), we thus obtain (Al3) J ljJl u l 2dx 2. f lui IOu.a.dljJldx. But dljJ = ECP' (EV )dV, so that lui IOu-a-dljJl2. Elcp'(EV)1 lui IOu·a·dVI ~ Elul (dV.a.dV)1/2(Ou.a_Ou)1/2
2. EluIW(K3 + K4Q+w2)1/2(ou.a.ou)1/2 2. (E/2)[(K 3 + K4q+w 2 ) lul 2 + W20u.a.OU]. In view of Proposition Al, we thus obtain from (Al3) f lJ!lul 2dx 2. EK~u~2, with K independent of E. Since ljJ t 1 as E + 0, we conclude that u = 0 as required.
(by
~u~
2
iii)
= 0, hence
The case m = 1_ In this case the function U satisfying (iv) is not needed. Since the only use made above of U was in the proof of Proposition Al, it
264
TOSIOKATO
suffices to prove the latter without using U. To this end we retrace the proof without using the cutoff function ¢ or, equivalently, setting ¢ = 1 and taking the integrals in (A9) on a finite interval (a,S) rather than (_00,00). The net result of this modification is to add the boundary terms (All) Re[W 2auDu]Ba = (1/2)[W 2a(dluI 2/dx)]Ba to the right of (A10), in which one should also set £ = 0 so that K7 , KS do not appear. Since u E L2(_00,00), there are sequences an ~ -00 and Sn ~ 00 such that dlu(x) 2dx is nonnegative for x = an and nonpositive for x = Sn' so that (All) 1
is nonpositive for a = an and B = Bn. Going to the limit (a,S) = (an,Sn) in (Ala) along such a sequence leads to the desired result.
-+
(-00,00)
Appendix 2. An example. As a simple example to which Theorem I applies, let arad(r) ~ (1 + r)P, q+(x).:: IxIO', q 0, p, 0' > o. Theorem I is applicable with W= 1, V = U = log(l + Ixl) provided P - 2:5.- (l-Tj)o. Since n can be arbitrarily small, it suffices that p < 2 + o. Thus any fast growth rate p for a rad is admissible if q+ grows fast enough. I do not know whether or not p = 2 + 0 is allowed, though it is all right if m = 1. Appendix 3. We sketch a proof that the assumptions in Theorem 1 of Evans [SJ imply (2.10). First we note that they imply, among other things, p:/2w' ~ K(l + 0~/2w), (A12 ) (r .:: 1) (Al3)
J
p-l/2(1 + 0~/2w)w dr
l
(A13) is exactly condition (iv) in [SJ, and (A12) follows directly from (i) there. Now (2.10) is obviously true if condition (v-a) of [8J is assumed. If, instead. (v-b) is assumed, then 01 is bounded. In this case suppose (2.10) is not true. Then it follows easily from the differential inequality (A12) that w is bounded, hence that the integral in (A13) is finite--a contradiction.
CORRECTIONS AND SUPPLEMENTS 1. In section 3.1, it was incorrectly implied that the result h max h. mln under assumptions (3.2) was due to Leinfelder-Simader [23]. Actually the same result had been given by Simon [29]. 2. The conjecture in Section 4.1 has been proved. 3. It has been shown that in Section 6, condition (6.5) can be weakened to (6.6).
SELE4DjOININESS PROBLLMS l'OR DIFFERENTI.4L OPERATORS
265
REFERENCES [1]
Browder, F. E., Functional analysis and partial differential equations, II, Math Ann. 145 (1962), 81-226.
[2]
Cordes, H. 0., Self-adjointness of powers of elliptic operators on noncompact manifolds, Math. Ann. 195 (1972), 257-272.
[3]
Devinatz, A., Essential self-adjointness of Schrodinger-type operators, J. Functional Anal. 25 (1977), 58-69.
[4]
Devinatz, A., Selfadjointness of second order degenerate-elliptic operators, Indiana Univ. Math. J. 27 (1978), 255-266.
[5]
Devinatz, A., On an inequality of Tosio Kato for degenerate-elliptic operators, J. Functional Anal. 32 (1979), 312-335.
[6J
Dung, N. X., Selfadjointness for higher-order elliptic operators, Dissertation, University of California, Berkeley, 1981.
[7J
Eastham, M. S. P., Evans, W. D., and McLeod, J. B., The essential selfadjointness of Schrodinger-type operators, Arch. Rational Mech. Anal. 60 (1976), 185-204.
[8J
Evans, W. D., On the essential self-adjointness of powers of Schrodingertype operators, Proc. Roy. Soc. Edinburgh 79A (1977), 61-77.
[9J
Evans, W. D., and Zettl, A., Dirichlet and separation results for Schrodinger-type operators, Proc. Roy. Soc. Edinburgh 80A (1978), 151-162.
[10] Everitt, W. N., and Giertz, M., Inequal ities and separation for Schrodinger type operators in L2(Rn), Proc. Roy. Soc. Edinburgh 79A (1978), 257-265. [11] Frehse, J., Essential selfadjointness of singular elliptic operators, Boletim da Soc. Brasil. de Mat. 8 (1977), 87-107. [12] Ikebe, T., and Kato, T., Uniqueness of the self-adjoint extension of singular elliptic differential operators, Arch. Rational Mech. Anal. 9 (1962), 77-92. [13] Jorgens, K., Wesentliche Selbstadjungiertheit singularer elliptischer Differentialoperatoren zweiter Ordnung in Co(G), Math. Scand. 15 (1964), 5-17. [14] Kalf, H., Gauss's theorem and the self-adjointness of Schrodinger operators, Arkiv. for Mat. 18 (1980), 19-47. [15] Kato, T., A second look at the essential selfadjointness of the Schrodinger operators, D. Reidal Pub. Co., Dordrecht 1974, 193-201. [16] Kato, T., Remarks on Schrodinger operators with vector potentials, Integral Equations and Operator Theory 1 (1978),103-113. [17] Kato, T., On the Cauchy problem for the (generalized) Korteweg-de Vries equation, to appear. [18] Keller, R. G., The essential self-adjointness of differential operators, Proc. Roy. Soc. Edinburgh 82A (1979), 305-344. [19] Keller, R. G., The essential self-adjointness of differential operators with positive coefficients, ibid. 345-360.
266
TOSIOKATO
[20] Knowles, r., On essential self-adjointness for singular elliptic differential operators, Math. Ann. 227 (1977), 155-172. [21] Knowles, I., On essential self-adjointness for Schrodinger operators with wildly oscillating potentials, J. Math. Anal. Appl. 66 (1978), 574-585. [22J Knowles, I., On the existence of minimal operators for Schrodinger-type differential expressions, Math. Ann. 233 (1978), 221-227. [23] Leinfelder, H., and Simader, C. G., Schrodinger operators with singular magnetic vector potentials, to appear. [24J 01einik, O. A., Linear equations of second order with nonnegative characteristic form, Mat. Sb. 69 (111) (1966),111-140; AMS Translation Ser. 2, vol. 65 (1967), 167-199. [25]
Phillips, R. S., Semi-groups of positive contraction operators, Czechoslovak. Math. J. 12 (87) (1962),294-313.
[26]
Read, T. T., A limit-point criterion for expressions with intermittently positive coefficients, J. London Math. Soc. (2) 15 (1977), 271-276.
[27]
Schechter, M., Essential self-adjointness of the Schrodinger operator with magnetic vector potential, J. Functional Anal. 20 (1975), 93-104.
[28]
Simon, B., Schrodinger operators with singular magnetic vector potentials, Math. Z. 131 (1973),361-370.
[29]
Simon, B., Maximal and minimal Schrodinger forms, J. Operator Theory 1 (1979), 37-47.
[30]
Sohr, H., Uber die Selbstadjungiertheit von Z. 160 (1978), 255-261 .
[31]
Sohr, H., Uber die Existenz von Wellenoperatoren f~r zeitabh~ngige Storungen, Monatsh. Math. 86 (1978), 63-81.
Schr~dinger-Operatoren,
..
* This work was partially supported by NSF Grant MCS-79-02578.
Math.
Spectral Theory of Differential Operators I.W. Knowles and R. T. Lewis leds.} © North-Holland Publishing Company, 1981
A WEYL THEORY FOR A CLASS OF ELLIPTIC BOUNDARY VALUE PROBLEMS ON A HALF-SPACE Robert M. Kauffman Department of Mathematics Western Washington University Bellingham, Washington, U.S.A.
A Weyl theory for a class of elliptic partial differential operators on a half-space is developed, and related to the well-posedness of the boundary value problem Mf
=
g, where f and g are square-
integrable on the half-space and f is required to satisfy certain conditions at the boundary.
The
operators discussed are natural generalizations of ordinary differential operators with positive coefficients.
Recent developments in the Weyl
theory for these ordinary differential operators are reviewed, and the ODE results are related to the PDE results. O.
INTRODUCTION
Consider the following problem:
"Solve the equation Mf
=
g on the
open region U, with g in L (U), where M is an elliptic partial dif2 We deal for simplicity with the case
ferential operator on U."
R~ = {xix = (xl"" ,xk ) and Xl > a}. We assume that the coefficients of M are restrictions to R~ of elements of COO (Rk), and are positive in a certain sense. We ask what con-
where U is the half-space
ditions must be imposed on f to make the problem well-posed. If one examines physical situations, such as the heat equation or the Schroedinger equation, where this problem arises, it becomes clear that one should expect to impose conditions at the boundary Xl = 0 and should also in many applied situations require f(x) to become small as Ixl becomes large. A very reasonable smallness condition is the condition that f be in
L2(R~).
into a problem in the Hilbert space L2(R~). stated problem in
This turns the problem Once one solves the
L2(R~), one may often use sernigroup theory to solve
related parabolic or hyperbolic problems such as
d2~/dt2
=
-M~. 267
3~/dt =
-M~
or
R.M. KA UFMANN
268
Having decided to impose these conditions on f, we must worry about two things.
First, we must ask whether the solution is uniquely
determined.
Second, we must ask whether we have imposed so many con-
ditions that, for some g, no solution exists.
The second worry turns
out to be groundless; there is always a solution when M and the boundary conditions at xl =
°
satisfy certain reasonable hypotheses.
However, even in situations where M and the boundary conditions appear very innocent, the solution may fail to be unique.
This un-
expected non-uniqueness occurs, when it occurs, because additional boundary conditions at infinity upon f are necessary to specify the solution; merely requiring f to be in
L2(R~)
is not enough.
We examine the question of boundary conditions at infinity both in the ODE and PDE case.
We relate the question to the essential self-
adjointness of a certain operator in L (R:), and examine conditions 2 on the coefficients to guarantee that the operator is essentially self-adjoint, and hence that the solution is unique.
This essential
self-adjointness is important in its own right because it means that in a certain sense the problem on the infinite region may be approximated by using finite regions. 1.
THE ONE-DIMENSIONAL CASE
In this section we examine the one-dimensional case, which is a pro~ lem in ordinary differential equations.
We change the region from
R! = (0,"') to (1,"') to make the statement of some of the theorems easier.
We first state the problem precisely.
Problem P.
Given g in L (1,"'), find an f in L 2 (1,00) such that Mf=g, 2 and such that (f(l), f(l) (1), .... ,f(2N-l)(1» is in S, where S is an 2N N-dimensional subspace of complex 2N-space K ,and where M is a 2Nth order differential expression.
We assume that M and S satisfy
the following: i) M = L~(-l)jDjPjDj, with D = djdx, and with each Pj the restriction to (1,"') of an element of C"'(-"',"'); ii) Pj ~ 0 on (1,"'); iii) PO ~ E > 0 on (1,"'); iv) PN > 0 on [1,"') where we have defined PN(l) by using the continuous extension of PN to [1,"'); 2N v) if f and g are in C [1,"'), with f(x) = g(x) x large, and with (f(l), f(l)(l), ... ,f(2N-l)(1» in Sand (g(l), g(l)(l), ... ,g(2N-l)(1»
in S, we assume that
JiMfg = L~ JiPjf(j)g(j)
0 for
269
Remark: Mf
=
To write M
=
L~(-l)jDjPjDj is to mean that
L~(-l)j(pjf(j))(j).
Remark:
Assumption v) above is a requirement upon S.
fied in many cases of interest for applications. satisfied if S is the set of all vectors v v.~ = 0 for i <- N - 1. Remark:
=(V
O
It is satis-
For example, it is
' ... ,v2N_~ such that
The following theorem is essentially due to Friedrichs.
Theorem 1.1. There exists a self-adjoint operator H in L (1,oo), such 2 that (Hf,f) ~ E(f,f) for all f ir. the domain of H, and such that, if f Proof:
H-lg, f is a solution to problem P. The operator H is just the Friedrichs extension of the oper-
ator R, where R is defined as follows:
domain R is the set of re-
strictions to (1,00) of elements g of C~(_oo,oo), such that (g(l) ,g(l)(l), ... ,g(2N-l)(1)) is in S, and Rf = Mf for any f in the domain of R.
It is not hard to see from the definition of the Fried-
richs extension and from the basic properties of differential operators in L 2 (1,00) that H has the required properties. Remark:
We now give a small sample of results which guarantee the
uniqueness of the solution to problem P.
Although an effort has
been made to make the sample representative, it is far too small to be complete.
Generally, only original results are quoted, rather
than some substantial improvements of these results which have been made later on.
Also, we only discuss positive coefficient versions
of results which are in many cases more general. We begin with a very interesting question. Question 1.2.
(Everitt [3], 1968).
Is the solution to problem P al-
ways unique? Remark: case.
The solution is well-known to be unique in the second-order In the higher order case, the question reduces to the defi-
ciency index problem for our operators, since for these operators the deficiency indices are both equal to the dimension of the squareintegrable solutions on [1, co) to the homogeneous equation Mf
= f).
It
may be shown by a simple dimension argument that the solution to problem P is non-unique if the dimension of the square-integrable solution space is more than N.
It is well-known from the theory of
ordinary differential operators that this dimension is at least N, and that the solution to problem P is unique if the dimension is N.
270
R.M. KAUFMANN
Theorem 1.3.
(Everitt [3], 1968).
Let M
solution to problem P is unique if PI
$
D4 - DplD + PO.
=
Then the
Kx 2 (PO + 1)1/2
Theorem 1.4. (Walker [7], 1971). The solution to problem P is unique for most values of a, ~ and I, where M = D2 x a D2 - Dx~D + Xl. Theorem 1.5. (Eastham [2], 1971).
Let M
solution to problem P is unique when PI
=
~
2
2
D P2D - DplD + PO. The 2 Kx , for some K > 0, and
Po is bounded. Theorem 1.6.
(Devinatz [1], 1973).
unique when M
=
2
D P2D
2
The solution to problem P is
- DplD + PO' where Po has bounded mean value,
and P2 is non-decreasing. Theorem 1.7. (Hinton [4], 1972). N
..
.
The solution to problem P is
unique when M = LO(-l)JDJpjDJ, with PN identically one, Pk = 0(t4k/4N-2) for k > 0, and Po arbitrary. Theorem 1.8.
(Kauffman [5], 1977).
Let M
=
i~(-l)jDjPjDj.
Suppose
each Pj is a finite sum of real multiples of real powers of x.
Then
the solution to problem P is unique when degree Pi - 2i hits its maximum value for only one i. exist examples when N
It is also unique when N
=
2.
There
3 of M where the solution to problem P is not unique; such examples may be found with M = _D 3 x a D3 + (:ix o - 6 , =
for certain values of a > 6 and (:i > O. Theorem 1.9.
(Kauffman [6], 1980).
Let M
=
Z~(-l)iDipiDi
Suppose
that the Pi satisfy certain regularity hypotheses, that PN ~ a > 0, and that, for some j, Pi = O(p~-a) for some a > 0 and all i j. J n Suppose in addition that for all n, x = O(p.). Then the solution
+
to problem P is unique. that p~j) l
=
J
One of the chief regularity hypotheses is
O(p~+1 + 1) for all positive I and j, although certain l
other technical hypotheses are necessary.
The regularity hypotheses
are satisfied, for example, if all Pi are finite sums of terms of the form fe g , where f and g are finite sums of real multiples of real powers of x. Remark:
The moral of theorems 1.8 and 1.9 seems to be that problem
P is well-posed if any coefficient may be regarded as the biggest, in the sense of these theorems, provided the coefficients are sufficiently regular. Remark:
Question 1.2 is still unresolved if N
=
2.
A related re-
sult of considerable interest is announced by T. T. Read in these Proceedings.
271
.1 It'/:YL THEORY
2.
THE PDE CASE
We now study the case of an elliptic partial differential operator k on the half space R+ {xixI > O}, where x = (xI,x2' ... ,xk ) is an element of Rk. Guided by the case of an ordinary differential operator, we I et M .
=
.
k
.
_.N
j
j
j
j
LIM i , wlth Mi - LOC-l) DiPijD i , where Di denotes N
...
.
.
dJ/dX~, and where by this notation, M.f = 2: (-1)J (lJ;aX~Cp .. C1Jf/dx~). 0 l l k l lJ l We assume that each p .. is the restriction to R+ of an element of k lJ C (R ), and that each Pij is non-negative, PiO ~ E > 0, and the con00
tinuous extension of each PiN to {xixI ~ O} is non-vanishing. Before we can phrase problem P, we need to deal with a new difficulty.
From the knowledge that f and Mf lie in L2CR~), one cannot
say anything about the behavior of f at the boundary of R~. (This is in sharp contrast to the ODE case.) Hence the imposition of boundary conditions is not possible unless we place more regularity requirements on f. Definition 2.1. We say that f is 2N-regular on R~ if, for all ¢ in Co CRk) , ¢f is in H2N CR~), where H2N CR~) is the Sobolev space of all functions g such that all partial derivatives of g of order up to and including 2N lie in L2(R~). in the distributional sense.) Remark:
(The partial derivatives are taken
The next lemma makes precise the notion of a boundary value.
Lemma 2.2. There exists a linear transformation T with domain the 2N-regular functions on R~ and with range contained in the set of ordered 2N-tuples of elements of L~ocCRk-l) such that, if Tf
=
ChO' ... ,h 2N - l ), the h. have the following properties: a) for any 8 in ~OCRk-I), 8h is in H2N-l-iCRk-I); i b) if f is in H2NCR~), then hi is in H2N-l-iCRk-l);
c) if fn is a sequence of 2N-regular functions on R~ such k that, for all ¢ in CO(R ), ¢f n converges to ¢f in H2NCR~), then, if Tf n (h nO ' h nl ,·· .hn2N - I ) and Tf = (h O ' ... ,h 2N - I ), it follows that, k-l for any 8 In CO(Rk-l ), 8 h . converges to Bh. in H2N-I-i CR); nl 2N k l d) if fn is a sequence in H CR+) and fn converges to f in H2N(R~), then Cusing the notation of part c) h ni converges to hi in •
00
H2N-I-iCRk-l) ; e) if f is the restriction to Rk of an element of C2N (Rk),
+
and Tf
=
(h O '.·· ,h 2N - I ), then
hiCx2'··· ,xk)
=
(li f / dx icO,x2'··· ,xk );
272
R.M. KAUFMANN
f) if Ch O"" ,h 2N _ l ) is an ordered 2N-tuple of complexvalued functions with each hi in C;CRk-l), then Ch "" ,h - ) = Tf O 2N l for some f which is the restriction to R~ of an element of CO(Rk ). Remark:
The content of the preceding lemma is that T is the continu-
ous extension, in a natural sense, of the restriction map defined in e).
The lemma is essentially well-known in the theory of PDE, al-
though a slight modification is needed to extend the usual trace mappings to the 2N-regular functions. Remark:
Now that we have defined what we mean by a boundary value,
we are ready to state problem P for the PDE case. Given g in L2(R~), find an f in L2(R~) such that f is k k 2N-regular on R+, Mf = g on R+, and Tf(x 2 , ... ,xk ) is in S for almost k l every (x "" ,x ) in R - , where S is an N-dimensional subspace of k 2 complex 2N-space K2N such that, for any f and g which are restric-
Problem P.
tions to
R~ of elements of C;CRk), with Tf(x 2 , ... ,xk) and
Tg(x , ... ,x ) in S for all points (x '" 2 k 2 N
j
.,X ) of Rk - l
k
j-
LO J k PljDlfDlg· R+ Notation 2.3.
Let W be the set of all f such that f is the restric-
tion to R! of an element of C;(Rk), with Tf(x 2 , ... ,x ) in S for all k k-l points (x2"" ,xk ) of R . Let R be the restriction of M to W. Let HR be the Friedrichs extension of R. Remark:
The following theorem is proved in Kauffman (to appear), al-
though it seems likely that a number of earlier writers, including Friedrichs, knew the result.
I felt it was necessary to give a
proof because I could not find an explicit reference. Theorem 2.4.
Every f in the domain of HR is 2N-regular, and
Tf(x , ... ,X ) is in S for almost every point (x 2 '" 2 k Hence, in particular, if f Furthermore, HRf = Mf.
k l k ) of R - . H-1 , f is a R
.,X =
solution to problem P. Remar~:
l~ HR =
R, where R is the operator theoretic closure of R,
then, for any f in the domain of HR , there is a sequence fn of elements of W such that fn converges to f and Mf n converges to HRf in L2(R~). This gives hope of computing things about HR by using compact support functions. Hence the question of when HR = R has some independent interest. Remark:
We now introduce two important properties.
273
A II'LTL l'HEORY
Property a.
The solution to problem P is unique.
Property b.
HR
Remark:
=
R.
(In other words, R is essentially self-adjoint.
We investigate the relationship between these two desirable
properties.
It follows from well-known theorems in ordinary dif-
ferential operator theory that they are equivalent in the ODE case. It is not hard to see that in the PDE case Property b implies Property a.
To go the other way, one first tries to study the or-
thogonal complement of range R.
Unfortunately, it is difficult to
find elements of this orthogonal complement which are regular enough to have boundary values.
Hence a more sophisticated argument seems
necessary. Theorem 2.5.
(Kauffman, to appear).
Let Rand H be as above.
Let Q
be the restriction of M to the set of 2N-regular f such that k-l Tf(x ,· .. x ) is in S for almost every point (x , ... ,X ) of R . 2 k 2 k These are equivalent: i) R is essentially self-adjoint; ii)
R=
iii) R
iv) HR
H , where R
R
is the operator-theoretic closure of R;
Q;
= =
Q;
v) Q is 1-1.
Furthermore, if R is not essentially self-adjoint, there exists an f such that Mf
=
0, f is in
L2(R~), all partial derivatives of f of all
orders are extendable to continuous functions on {x I xl :: O}, and (f, Dlf, ... ,DiN-If)
(0,x " , .xk ) is in S at all points (0,x 2, ... ,xk) 2 of the hyperplane xl = 0, where we have defined these partial derivatives at xl Remark:
=
0 by using their continuous extensions.
Property a is the same as Property v) of the theorem, and
Property b is the same as Propert i).
Hence Properties a and bare
equivalent. Proof of Theorem 2.5:
We give a brief sketch of the proof of theo-
rem.
R is contained in HR , it is clear that i) implies ii). Since integration by parts may be used to show that Q is contained in K",
Since
it is clear that ii) implies iii).
Since R is contained in H , and R HR is contained in Q, it follows that iii) implies iv). Since HR is 1-1, it is clear that iv) implies v). We now prove the only hard part of the theorem; we show that v) implies i).
Let F be the Friedrichs extension of R2.
It is possible,
274
R.M. KAUFMANN
with some effort, to prove that for any f in the domain of F, f is 4N-regular, and T(Mf)(x 2 , ... xk) is in S for almost every (x2"" ,xk) of Rk - l From the definition of the Friedrichs extension, it is clear that domain F is contained in domain
R.
If R is onto, it is self-adjoint, since it is symmetric.
If R is
not onto, then, since range R is closed, there is an element ¢ of COO (Rk) such that ¢ is not in the range of R. ¢ is clearly in vI. But,
o +
since F is onto, M¢ = Ff for some f. Q(¢ - Rf) to.
O.
=
But ¢ - Rf
f O.
Hence M¢ = Q¢ = Q(Rf).
Thus
Hence Q is not 1-1, if R is not on-
The proof of the equivalence of i) -v) is completed.
be shown that f is m-regular for all positive m.
It may
One may use this
fact together with Sobolev's imbedding theorem to prove the final assertion. Question.
What are conditions on the coefficients which guarantee
that R is essentially self-adjoint? Remark:
We answer the question for certain types of coefficients.
Our results apply to the whole-space case as well as the half space case, and are new for the half-space and higher-order whole-space cases.
In the whole-space case, we let R be the restriction of M to Our theorem
C~(Rk), and ask whether R is essentially self-adjoint.
contains no new assertions about the second-order whole-space case, as the specialization of our result to this case follows as a very special case of the strong second-order theorem announced by T. Kato in these Proceedings. Remark:
Our results are about coefficients which are like polynomi-
als, but are more general.
The virtue of this more general class is
that it permits arbitrary exponents and is translation-invariant. Definition 2.6. 2 f(x) c¢(x) (x
We say that f is in Z[a,oo) if
+ 1)A/2 + ~(x) + y(x), where c is a complex number
and i) ¢ and
~
are restrictions to [a,"') of elements of
ii) y is the restriction to [a,"') of an element of C~(-oo,oo);
iii) ¢(x) approaches 1 as x approaches infinity; (;) 2 -·/2 f or all j ::: 1; i v) ¢ J (x) = 0 (x + 1) J 2 v) ~(j)(x) = o(x + 1)(A-j)/2 for all j ~ 0; vi)
~
=
0 if c
=
O.
A is called the degree of f.
We take A
-00
if c
O.
275
A II'E1'1- l'HH)R Y
Definition 2.7. be in
z(-oo,~)
A complex-valued function f in Coo(_oo,oo) is said to
if
i) the restriction of f to [0,00) is in Z[O,oo); ii) if g(x)
f(-x), the restriction of g to [0,00) is in
=
Z[O,oo). Theorem 2.S. N
Mi
=
(Kauffman, to appear).
.,
.
~O(-I)JDfPijDf'
Let M
=
~~i'
where
Let R be as in Notation 2.3.
Assume the
following: i) for i > 1 and all j, Pij(x) = hijCx i ) for all x in k R+, where h .. is in Z(-oo,oo);
lJ
ii) Plj(x) = hlj(x l ), where h lj is in Z[O,oo); iii) if n(l,i,j) is the degree of the restriction of h ij to [0,00) for i ~ 1, and if n(2,i,j) is the degree of the restriction of the function gij(x i ) = hij(-x i ) to [0,00) for all i > 1, then n(l,i,j) 2j < n(l,i,O) for all i ~ 1 and all j > 0, and n(2,i,j)
2j < n(2,i,0) for all i > 1 and all j
°
> 0.
°
iv) Pij ~ for all i and j, PiO ~ E ~ for all i, and the continuous extension of PiN to Xl 2 is non-vanishing for all i.
°
Then R is essentially self-adjoint Remark:
It should be noted that Pij must be a "polynomial" in Xi
only, by hypotheses i) and ii). Remar~:
In the ODE case, any coefficient is allowed to be the big-
gest, where the size is measured by taking degree Pj - 2j.
In the
PDE case, we need PiO to be the biggest, using this measure of size. k
Theorem 2.8'.
Let M
N
..
.
LIM i , where Mi LO(-I)JDIPijDi. k Suppose each PiJ' is in Coo(R ). Let R be the restriction of M to k CO(R ). Assume the followin?: =
00
i) for all i and j, Pij (x)
(Xi) , with h ij in Z (-co, 00) ; the restriction of h ij to [0,00), and n(2,i,j) is the degree of the restriction of the =
ii) if nCI,i,j) is the degree
h
ij
of
func~ion gij(xi) = hij(-xi) to [0,"'), then n(l,i,j) - 2j and n(2,i,j) - 2j < n(Z,i,O) for all i and all j > 0;
°
iii) Pij ~ for alJ. i and j, PiO 2 PiN is non-vanishing for all i.
E
>
°
<
n(l,i,O)
for all i, and
Then R is essentially self-adjoint Remark:
We now discuss examples where R is not essentially self-
adjoint.
To do this, we review a few concepts from ordinary dif-
ferential operator theory.
276
R.M. KAUf-MANN
Definition 2.9. and PN >
°
Let L
N iDi PiD i ,wlt · h eac h Pi LO(-l)
=
on the interval [a,oo).
~
0, Po
~
E
> 0,
Suppose each Pi is the restric-
tion to [a,m) of an element of eW(_oo,oo). Then L is said to be limit-N on [a,oo) if there exist exactly N linearly independent solutions to Lf =
° in
L [u,oo). 2
A parallel defi-
nition applies to L on (-oo,aJ. Remarks:
It is well-known that for any L in the above definition,
there exists at least N linearly independent L [a,oo) solutions to 2 = 0. The same result holds on (-ro,aJ. Hence L can fail to be
Lf
limit-N on (-oo,aJ or [a,oo) only by having N independent square-integrable solutions.
+
1 or more linearly
It is also well-known that
if b > a, L is limit-N on [a,oo) if and only if L is limit-N on [b,oo); if b < a, L is limit-N on (-oo,aJ if and only if L is limit-N on (-oo,bJ.
Finally, it is well-known that, if L
=
L~(-l)jDjPjDj,
with each Pj in eoo(_oo,m), Pj ~ 0, PN > 0, and Po ~ E > 0, and L is not limit-N on some interval [a,oo) or (-oo,aJ, there is a non-trivial f such that Lf
=
°
and f is in L 2 (-00,00).
Remark:
Recall that, as discussed in section 1 it is shown in a 6 Kauffman [5J that there exist L of the form L =_D 3 x a D3 + bx - , with a > 6 and b > 0, such that L is not limit-3 on [1,00). Remark:
If L is limit-N in the sense of our definition, it is not
hard to show that the deficiency indices of the minimal operator corresponding to L on [a,oo) are both equal to N, and conversely. Hence our definition is equivalent to the usual definition Theorem 2.10. N
..
(Kauffman, to appear). .
Suppose M
=
L~., with l
Mi = LO(-l)JDIPijDI' Suppose Pij(x) = hij(xi)' where h ij is in eOO(_oo,oo) for i > 1, and hI. is the restriction to [0,00) of an ele00 J ment of e (_00,00). Suppose that hlN is non-vanishing on [0,00), and h iN is non-vanishing on (_00,00) for i > 1. each i, and each Pij is non-negative.
Suppose PiO
~
E >
° for
Let R be as defined above.
Let L· = LNO(-l)jDjh .. Dj. Then, if R is essentially self-adjoint, Ll l lJ is limit-N on [0,00), and Li is limit-N on (-00,0] and [0,00) for i > 1. Remark:
Although the examples given above are of Li which are not
limit-N on [1,00), for N
=
3, it is easy to extend these expressions
to expressions on [0,00), which can not be limit-3 by the above remarks. Proof of Theorem 2.10:
It is well-known that if Li is not limit-N
on [0,00) or (-oo,OJ for some i > 1, there is a non-trivial solution
277
.4 IVnYL THEORY
to Li f = 0 such that f is in L2 (_00,00) .
If Ll is not limit-N on
[0,00), then there is a non-trivial f such that Llf
=
°
on [0,00) and
f is in L [0,00), and such that (f(0),f(l)(0), ... ,f(2N-l)(0» 2
is in
S.
Select any i
such that Li is not limit-N on some half-line, and let
f be the square-integrable solution constructed above. k = f(xi) for x in R+. Note that Mifi = 0.
Define
fi(x)
Pick any ¢ in C~(_oo,OO) such that ¢ is identically one on a neighbor-
°
hood of zero, and such that ¢(y) = for lyl ~ 1. Let 6 j (x) = ¢(x j ) k for j ~ 1, and for any x in R+. Let g TI j i 6j f i . k o o k Note that Mig = and g is in L 2 (R+). Note that g is in C (R+) and all partial derivatives of g are extendable to continuous functions k-l on {xl Xl 20}. Note that Tg(z) is in S at any point z of R
°
+
k For any j, Mjg is in L 2 (R+). Hence g is in the domain of Q, where is defined in Theorem 2.5. By Theorem 2.5, if R is essentially
self-adjoint,
R
Q.
=
in the domain of
Hence, if R is essentially self-adjoint, g is
R.
It is not hard to see, however, that for any g in the domcin of (Mig,g)
~
c(g,g).
self-adjoint.
3.
Q
R,
This is a contradiction, if R is essentially
The theorem is proved.
UNANSWERED QUESTIONS
In conclusion, it seems worthwhile to list some interesting problems which have not yet been solved.
2
2
Problem 1. Let L D P2D - DplD + PO' with each Pi ;:: 0, PN > 0, and p·O ~ E > 0 on [a,m). Suppose each Pi is the restriction to
Is L necessarily limit-2?
[a,m) of an element of C"'(-oo,oo).
(Equivalently, is problem P well-posed for L?) Problem 2.
Let L
N
"
.
;:: 0, PN > 0, and j Po ;:: c > 0 on [a,"'). Suppose each Pj is the restriction to [a,"') of an element of CeD(_ro,w). Is it possible for all solutions to Lf = 0 =
LO(-l)JDJPjDJ, where each P
to be in L 2 [a,oo)? Problem 3.
Let M and Li be as in Theorem 2.10.
limit-N on each half-line.
Suppose each Li is
Is R necessarily essentially self-
adjoint? Problem 4.
Let M be as in section 2.
Can there exist two N2N dimensional subspaces Sl and S2 of complex 2N-space K ,such that
Sl and S2 are as discussed in the definition of problem P, and such
278
R.M. KAUFMANN
that problem P is well posed for Sl and not for S2? REFERENCES [1] Devinatz, A., Positive definite fourth order differential operators, J. London Math. Soc.
(2) 6 (1973), 412-16.
[2] Eastham, M.S.P., The limit-2 case of fourth order differential equations, Quart. J. Math.
Oxford (2) 22 (1971), 131-34.
[3] Everitt, W.N., Some positive definite differential operators, J. London Math. Soc.
(1) 43 (1968), 465-73.
[4] Hinton, D.B., Limit-point criteria for differential equations, Canad. J. Math. 24 (1972), 293-305. [5] Kauffman, R.M., On the limit-n classification of ordinary differential operators with positive coefficients, Proc. London Math. Soc.
(3)
35 (1977), 496-526.
[6] Kauffman, R.M., On the limit-n classification of ordinary differential operators with positive coefficients (II), Proc. London Math. Soc. (3) 41 (1980), 499-515. [7] Walker, P.W., Deficiency indices of fourth-order singular differential operators, J. Diff. Eq. 9 (1971), 133-41.
Spectral Theory of Differential Operators I.W. Knowles and R. T. Lewis (eds.) ©North-Hol/and Publishing Company, 1981
ON THE CORRECTNESS OF BOUNDARY CONDITIONS FOR CERTAIN LINEAR DIFFERENTIAL OPERATORS Ian Knowl es Department of Mathematics University of Alabama in Birmingham Birmingham, AL 35294 David Race Department of Mathematics University of the Witwatersrand Johannesburg 2001 South Africa
For ordinary linear differential expressions T of order 2n defined on a real interval 1, the problem of determining which linear homogeneous boundary conditions give rise to well-posed differential operators in L2 (I) is considered. For the case I = [0,00), it is shown that all the operators obtained by imposing n linearly independent (complex) boundary conditions at 0 are well-posed, under appropriate conditions on the coefficients of T. The regular case, 1= [O,lJ, is also discussed. The problem of correctly assigning boundary conditions to formal differential expressions arising from physical models, and elsewhere, is of central importance in applications of differential operator theory. For definiteness, consider the differential expression T defined by TY(X) = (_l)n y(2n) + nil (Pn_r(x)y(r)) (r), X E I, (1 ) r=O where I c R and the coefficients Pi('), 1 ~ i ~ n, are complex-valued and locally Lebesque integrable on I, We associate with T the usual maximal and minimal operators, Tl and TO respectively, in L2(I) as follows (see [12J): Let f[iJ denote the ith quasi-derivative of a function f (see [12, p. 49J). The operator Tl is then given by D(Tl)={fE L2(I): f[i], 0 ~ i ::c 2n-1, are locally absolutely continuous, and Tf E L2(I)} Tlf = Tf, f E D(T l ), while TO is defined to be the closure of the operator TO given by 279
IAN W. KNOWLES and D. RACE
280
OtTO)
{f EO(T l ): f vanishes outside some compact interval [a,S] TOf = d,
f
E
C
(1)}
OtTO)'
The essence of the problem of assigning boundary conditions is roughly the following: one must choose the boundary conditions so that the associated restriction, T, of Tl has domain optimally large in some suitable sense. In the best of cases this means that the spectrum of the operator T allows something like an eigenfunction expansion theory. If there are too few boundary conditions, one can expect the point spectrum of T to fill out the complex plane; and if there are too many boundary conditions the point spectrum of the adjoint operator may do likewise (in which case the residual spectrum of T may cover the complex plane). Clearly, a minimum requirement on the spectrum of T is that the resolvent set, p(T), be non-empty. The following necessary condition for this to occur forms a convenient starting point for our discussion: Lemma [2, p. 1311]. Let T be an operator obtained from T by imposing a (possibly empty) set of boundary conditions on O(T l ), and let A E p(T). Then the number of linearly independent boundary conditions defining T is equal to the number of linearly independent solutions of the equation TY = AY that belong to L2(I). Our main concern here is to investigate the converse result; i.e., to determine conditions under which an operator T, obtained from T by imposing the number of boundary conditions specified in the theorem, has non-empty resolvent. That the converse is not true in general may be seen from the following example: Let T be defined by OtT) = {f E L2[0,1]: f' is absolutely continuous, f" E L2[0,1], and f(O) + f(l) = f'(O) - fl(l) = O} Tf(x) = -f"(x), 02- x 2- 1, f E otT). Here, the associated characteristic polynomial for the eigenvalues of T is identically zero, giving Po(T) = [, where Po(T) denotes the point spectrum of T. Clearly it is of interest to document precisely when such pathological cases occur, as one would expect, among other things, that any attempt at a numerical solution of a boundary value problem involving such an operator, would fail. On the other hand, it should be noted that even if one knows that p(T) is not empty, the associated spectral theory can still be extremely complicated ([1l,§5.4; 10;
9]). In the sequel, we denote the regularity field of an operator T, TO eTc Tl , by n(T); the essential spectrum of T is denoted by Eo(T), and the residual spectrum by Ru(T) (c.f. [8,§2]). An extension T of TO is called well-posed if n(T) is not empty. It is known (see [8,§3]) that T is non-well-posed if and only if n(T O) C PutT).
281
BOUND,4R Y CONDITIONS FOR DUTERENTIAL OPERA TORS
We consider firstly the so-called regular case in which we take I ~ [O,lJ for simplicity, and assume Pi(·) EO L[O,lJ, 1 ~ i 2. n. Notice that n(TO) = [ , and thus an extension T of TO is non-well-posed if and only if Po(T) = [. Given matrices A = (a rs ) and B = (b rs ) of order 2n and with complex entries, define the operator TAB by V(T AB ) = {f EO L2[0,lJ: fCiJ, 0 < i < 2n-l, are locally absolutely conti~uou~, Tf EO L2[O,lJ, and
:~~
a rs f[s-lJ (0) + brs f[s-l] (1)
=
0 for
~r~
2n} (2)
For general n, rather little is known about which extensions TAB are well-posed. One can reduce the problem to the case p.1 (.) = 0, 1 -< i -< n, by means of known asymptotic formulae for the solutions y(X,A) of (T - A)y = 0 valid for fixed x and IAI ~ 00 (c.f. [11,12J). For separated boundary conditions the extensions TAB are always well-posed ([11, Lemma 3, p. 94J). For n = 1 it is not difficult to show that TAB is non-well-posed if and only if a12
a 21
b ll b 21
all
a12
a 21
a 22
all
+
a 22
b12 b22
0,
bll b 21
b 12 b22
0,
and
= o. In particular it follows directly that TAB is well-posed whenever the boundary conditions are J-selfadjoint (where J denotes complex conjugation in L2[0,1]; see It seems unlikely that for general n all J-selfadjoint operators TAB [8,13J). are well-posed, although there are no examples confirming this, as yet. n
x
We now concentrate on the singular case, I = [0,00). Let A = (a rs ) be an 2n matrix of complex numbers with rank n. Define the extension TA of TO by V(T ) A
~
{f
E
D(T ): 2f a f[s-lJ (0) = 0, l s=l rs
r=1,2,···,n}
(3) TA f = Tf, f EO V(T A) Then we have Theorem 1. If Pi = 0, 1 < i 2. n, then for every choice of the boundary matrix A, the extension TA of TO is well-posed.
Proof. Observe that by [5, p. 106], we have Eo(Ta) = [0,00), and hence that rr(T O) = C - [0,00); it is thus sufficient to determine when a complex number
282
IAN W. KNOWLHS and D. RACE
A ¢. a: - [0,00) lies in Po(TA)'
For such a A consider, then, the equation, (- 1 )n f (2n) = Af.
(4 )
Let p denote the 2nth root of (-1 )nA satisfying TI/2 < arg p < n/2 + n/n. Then . 2n, were h the distinct 2n th roots of (-1 )n A are given by lJi =- PE i - 1 , 1 :::.- 1:::'£ = exp(iTI/n). Any eigenfunction of TA must be of the form f(x) = c exp(px) + c 2 exp(P£x) + ... + c n exp(P£ n-l xl (5) l for appropriate constants c ,,·· ,c ' Using (5), one can show that the characterl n istic equation for the eigenvalues of TA has the form 6(A) = det(M) = 0, where M = (m ij ) and ((j-l))S-l _ 2n I a is p£ mij(p ) - s=l . -1 i-l We can write M = AG, where G = (g .. ) the 2n x n matrix with g .. = ( PE J ) in th . th 1J 1J the i row and j column. In this case the formula for the determinant of the product is
where A denotes the n x n matrix consisting of columns sl ,s2,···,sn of A, sl ... sn and Gsl ... sn denotes the n x n matrix consisting of rows sl ,s2"" ,sn of G. The equat10n for the eigenvalues of TA thus becomes sl+s2+···+ s n- n si- l
I
P
( IT
1':-:h<s2<· .. <sn:::.- 2n
(7)
(£
i>j
Clearly, TA is well-posed if and only if equation (7) is not identically zero. As A is of rank n, at least one of the determinants det As ... s is not zero. 1
n
: det A t O}. Then, by [11, Lemma 2, p. 91] n sl' .. sn the term in ps-n in (7) is non-trivial, and the result follows.
Let s
=
max{sl + s2 + ... + S
#
Remark. This result is of independent interest. In much of the qualitative spectral theory of non-selfadjoint differential operators, one is forced (at least implicitly) to make artificial assumptions in order to exclude the "bad" extensions (see e.g. [1, p. 11, 9,. 15; 4; 7, §9]). It is therefore very useful to know precisely when such extensions cannot occur. Provided the coefficients Pi' 1 :::.- i :::.- n, are not too large, Theorem 1 can be extended to cover more general operators c. More precisely we have Theorem 2. If Pi = qi + r i , where qi E Lm[O,m), and r i E L[O,m), 1 :::.- i ~ n, then the extension TA of To defined by (3) is well-posed for every choice of the boundary matrix A.
m.
Remark. This includes the case Pi E L 1[0,00), 1 < m < 00, 1 ~ i ~ n, as one can i always write an r-integrable function, 1 < r < 00, as the sum of an integrable function and a bounded function.
283
IlOUNDAR Y COIVDITlONS FOR U1I'H'RENnAL OPERATORS
Proof.
The proof is divided into several stages.
the equation TY
Following [12,§22.2] we write
J..y in system form as
=
dY = A(x) Y dx (y,y[l], ... ,y[2n-1 J )T, and A(x)
=
AO(x) + Al(x) where AO and
Al are defined in [12, p. 176J, and we have set Po
=
1.
where Y = Y(x,r)
=
(8)
Let
Y = BU
( 9)
where
B
III
1l2n
n III
n 1l2n
n+l -Ill
-11
(_ 1 ) n- lll~n- 1
(10)
n+l 2n
(_ 1 ) n- \~~- 1
and Ili' 1.::. i.::. 2n, are the distinct 2nth roots of (-l)nJ,. defined earlier. Set B- 1 = (B ij ) and define A2 and A3 to be the matrices obtained from Al by replacing the elements p,., 1 < i ~ n, by q. and r., respectively. The system (8) then
,
,
becomes (11 )
(12 )
It is not hard to see that the functions c ij (x,·) and f ij (x,·) are analytic for fixed x> O. Also for Ipl ?:- 1 we have ICij(x,p)1 ~Kn
n
I
k=l
(13)
Iqk(x)l; Ifij(x,p)I.::.L n
where Kn and Ln are independent of x and p. Our initial goal is to determine the asymptotics of certain solutions of (11), from which we easily obtain the behaviour of solutions of TY = J..Y via (8). Before doing this we digress for a moment. H
=
C-
(f
x
o and consider, for
<
i
<
-
F) W + W(J
x
Define H = (h ij ) by
F) - (j
F) W(J
0
0
0
n the solutions V. ,
x
=
x
F)
(v .. ) of the integral equations
"J
284
IAN W. KNOWLES and D. RACE
]..I.x
X]..l. (X-i;) 2n e 1 2: hik (t:,p)v.k(i;,p)di; o k=l 1 X]..l.(x-t;)2n viJ·(x,p) J eJ L hjk(i;,p)vik(t;,p)dt; if j 'f i, .::. j < n (14) o k=l 00 ]..I.(x-t;) 2n vij(x,p) = - J e J L h' (t;,p)v.k(t;,p)di,; if n + 1.::. j < 2n. X k= 1 J k 1
Vii (x,p) = e
1
+
J
Observe that the solutions Vi'
.::. i .::. n, all satisfy the equation
~~ = (W + H)V. (11)' We now adapt the techniques of Kamimura [7] to show that the solutions Vi = Vi(x,p), 1 .::. i .::. n, of (14) exist and have components square integrable in [0,00), for suitable values of p. To do this we require analogues of certain inequal ities used in [7]. Suppose that y E L2[0,00), and that ]..I is a complex number whose real part, y, is positive. Then (15) X
X
(16)
JOO I
JX e-]..I(x-t) y(t)dtI 2 dx.::. y-2 foo ly(t)I L dt
o
0
(17)
0
The first of these, (15), is a direct consequence of the Cauchy-Schwarz inequalit~ The other inequalities may be obtained from Young's inequality (see, for example, [15, p. 32]); using the notation of [15], one obtains (16) by setting f(t)
=
e]..lt,
t < 0
o ,
t > 0
and g(t)
= y(t),
t > 0 , t
together with r = q function
h(t) = e-]..It,
=0
,
t
>
0
t < O.
Let L denote the Cartesian product of 2n copies of L2[O,00); with the usual inner product topology, Lis a Hi 1bert space. Defi ne S: L -+ L by x]..l.(x-t;)2n J e l L h.k(t;)fk(t;)dt;, < i < n o k=l 1 (Sf)i (x) 00 ]..I. (x-t;) 2n J e l L hik(t;)fk(t;)dt;, n + 1 < < 2n x k=l
285
BOUNDAR Y CONDITIONS ['OR DII'HiREN1'lAL OPERATORS
where f = (f ,···,f2n) E L. Using (13), (16), and (17), one can l show that 5 is a bounded linear operator in L with 11511 ~ G(»)
=
y
-1
(18)
an
where an does not depend on » and y =
min{IRe )J.I: 1 1
(we assume that -1T
<
< -
arg A ~ 1T).
as a bounded operator on L.
i
2nl
< -
1»l l / 2n Isin(~)1 2n
=
Thus for» such that G(A)
<
(19)
1, (I -
sf 1
exists
If we now write the integral equations (14) in the
form (I - S)V i = Ei where Ei = exp()Jix)ei (e denoting the i th standard basis i vector in R2n ), then it is clear that for each i, 1 ~ i ~ n, (14) has a unique solution Vi(·,p) E L for all» such that G(»)
<
1.
These solutions are linearly
independent, and for each fixed i the components, v ij ' 1 (c.f. [7]) lim v.J.(O,p) G(» )+0 1 Consider now equation (11). 1
=j
if
., j.
j
~
2n, of Vi satisfy
(20)
=
0
For 1
U.(x,p) = (I +
if
~
<
f
i
<
n, set
x
F(t)dt)V.(x,p)
0
(21)
1
One can show directly that the vectors U.(·,p) E L form a set of n linearly inde1
pendent solutions of (11). 1
<
-
j
<
-
In addition, it is clear that the components, uij ' 2n, of U.1 also satisfy (20).
Finally, consider the solutions of the equation TY = »y for» E [ - [0,00). By modifying [5, Theorem 9, p. 138J along the lines of [14, Theorem 3.2J one can show that Eo(T ) = Eo(T A) = [0,00), and hence that 1T(T ) = [ - [0,00). FurtherO O more, it is known ([6, Theorem C and equation (2.8)J) that for any» E 1T(TO) there are precisely n square integrable solutions of the equation TY = »y. fine y.1 = (Y'J') = BU., 1 1 1 (21).
Set Yk
=
i
<
-
Ykl for k
=
<
-
De-
n, where U.1 are the solutions of (11) defined by
1,2,··· ,n.
Then the functions Yk' k
=
1,2,· .. ,n,
constitute n linearly independent square integrable solutions of the equation TY = Ay; the quasi-derivatives y~j-1], 1 ~ j ~ 2n, are given by y~j-1J = Ykj· Also, by (9), (10), (20) we have for 1
~
k
~
n,
Yk(O,P) = 1 + 0(1) [1] Yk (O,p) )Jk[l + o(l)J
y~nJ(O,p) Yk[n+1J(0 ,p )
n-l
)Jk
[1 + o(l)J
-)J~[l + o(l)J
y~2n-1J(O,p) = (_l)n-l)J~n-l[l + o(l)J as G(A)
+
0, where G(A) is defined by (18).
(22)
286
IAN W. KNOWLES and D. RAC1!
We can now complete the proof of the theorem. Arguing as in the proof of Theorem 1, one can deduce that the eigenvalues A of TA are given by the roots of O(A) = 0, where 2n 2n [s- 1J a l sY n (0, p) I a lsYl[s- 1J (O,p) I s=l s=l O(Ic) 2n
I s=l
2n ansyfs-l](O,p)
I
s=l
a Y[s- 1J (0 ) ns n ,p
Using (22) we then have that O(A) = 6(A)[1 + o(l)J as G(A) -)- a where 6(A) is the corresponding determinant in the proof of Theorem 1. As 6(A) is never identically zero, it follows that O(A) has the same property, for any choice of the boundary matrix A. #
Remarks. 1. One can deduce from the proof of Theorem 2 that the eigenvalues of TA are enclosed by a curve in the A-plane of the form G(A) = constant. For example, when n = 1 and A = r exp(ie), the curve is of the form r'''lsin(e/2) 1 = k. This curve encloses the entire non-negative half-axis, and is in fact asymptotic to it for arg A approaching a or 211. This behaviour is consistent with the fact that there are known examples (see e.g. [3J), in the second order case, of operators with Lr[O,oo)-coefficients (r > 1) having an unbounded set of positive eigenvalues. If q. = 0, 1 < i < n, the appropriate asymptotic formulae for the solution values y:[j-1J(0,~) a;e valid for IAI -)- 00, and the point spectrum is thereby confined to some disc. 2. One can infer rather more about the spectra of the operators TA. Firstly, the point spectrum is always discrete in a: - [0,00), and has all its 1imit-points on [0,00). Furthermore, the eigenvalue equation for TA has the same general form as that for TA. Consequently, similar remarks to the above apply to Ro(T A). Thus, in general Ro(T A) U Po(T A) is discrete and confined to a neighbourhood of the non-negative real axis in the A-plane; i.e., a: - [0,00) c p(T A), with the possible exception of a discrete set. 3. Finally, we observe that this theory may also be extended to cover formal differential expressions of the form T1Y(x) = (_1)n y (2n) + 2nil Pr(x)y(r), a < x < 00. r=O In this case one can obtain similar results by combining Theorem 1 with the asymptotic formulae for solutions of T1Y = Ay given in [1,§2].
BOUNDAR Y CONDl'llONS FUR Dlt+f:REN'lL1L OPERATURS
287
REFERENCES [1]
Chyong, F. van, On one condition of finiteness of the set of eigenvalues for a non-selfadjoint ordinary differential operator of higher order (Russian), Vestni k Mosk. Univ. 21 (No.3) (1966), 3-13.
[2J
Dunford, N. and Schwartz, J. T., Linear Operators, Volume II, Interscience, 1963.
[3]
Eastham, M. S. P. and McLeod, J. B., The existence of eigenvalues embedded in the continuous spectrum of ordinary differential operators, MRC Technical Report 1688, Madison, Wisconsin, 1976.
[4]
Gimadlislamov, M. G., On an eigenfunction expansion of a non-selfadjoint differential operator of even order in a space of vector functions (Russian), Dokl. Akad. Nauk SSSR 143(1962),13-16.
[5J
Glazman, I. M., Direct Methods of Qualitative Spectral Analysis of Singular Differential Operators, IPST, Jerusalem, 1965.
[6]
Hinton, D. B., Strong limit-point and Dirichlet criteria for ordinary differential expressions of order 2n, Proc. Royal Soc. Edinburgh 76A(1977), 301-310.
[7]
Kamimura, Y., On the spectrum of an ordinary differential operator with an r-inte9rable complex-valued potential, J. Lond. Math. Soc. (2), 20(1979), 86-100.
[8J
Knowles, I., On the boundary conditions characterizing J-selfadjoint extensions of J-symmetric operators, J. Differential Equations, 39(1981).
[9J
Lidskii, V. B., Summability of series in terms of the principal vectors of non-selfadjoint operators, Amer. Math. Soc. Transl. (2) 40, 193-228.
[lOJ Naimark, M. A., Investigation of the spectrum and the expansion in eigenfunctions of a non-selfadjoint differential operator of the second order on a semi-axis, Amer. Math. Soc. Transl 16 (1960), 103-193. [llJ Naimark, M. A., Linear Differential Operators, Volume I, Ungar, New York, 1967. [12J Naimark, M. A., Linear Differential Operators, Volume II, Ungar, New York, 1968. [13J Race, D., The spectral theory of complex Sturm-Liouville operators, Ph.D. Thesis, University of the Witwatersrand, Johannesburg, 1980. [14J Race, D., On the location of the essential spectra and regularity fields of complex Sturm-Liouville operators, Proc. Royal Soc. Edinburgh 85A(1980),1-14, [15J Reed, M. and Simon, B., Methods of Modern Mathematical Physics, Volume II, Academic Press, New York, 1975.
This page intentionally left blank
This Page Intentionally Left Blank
Spectral Theory of Differential Operators I.W Knowles and R. T. Lewis (eds.) © North·Holland Publishing Company, 1981
INDEX AND NONHOMOGENEOUS CONDITIONS FOR LINEAR MANIFOLDS SUNG J. LEE DEPARTMENT OF MATHEMATICS PAN AMERICAN UNIVERSITY EDINBURG, TEXAS 78539 U.S.A.
A general formula is given for the index of a linear manifold, which is expressed by the possibly infinite dimensions of linear manifolds. A necessary and sufficient condition is given for a boundary value problem subject to infinite nonhomogeneous boundary conditions to have a solution. The theory is developed with the view that there are a minimal subspace and a maximal subspace and any intermediate subspace is the kernel of finite or infinite linear equations. It is motivated from ordinary differential operators whose solutions are allowed to be piecewise continuous at infinitely many points.
INTRODUCTION Let To C Tl be closed linear manifolds in the direct sum, Xl f!) X2 , of complex Banach spaces Xl' X2 such that the quotient space Tl/To is isomorphic to 12 and the null spaces of Tl and T; (see [2J for definition) are isomorphic to separable Hilbert spaces. Here 12 is the Hilbert space 1 x complex matrices a with aa* < Since Tl/To is isomorphic to £2' it follows ([7], [6J) that there exists a continuous linear operator B from T1 onto £2 whose kernel is To, and there exists a w*-continuous linear operator S+ from T; onto £2 whose kernel is T~. Moreover, there exists a unique x nonsingular Hllbert matrix C (see [3J, [5J for definition and some consequences) depending only on Band B+ such that 00
00.
00
00
(1)
for all a = {aI'a 2 }c; Tp b = {b 2 ,bj}
c;
* To.
A m x Hilbert matrix P is called a normalized one if it satisfies the condition: if m < then the rows P are linearly independent in 10' and if m = then all the rows of Pare orthonormalized in 1 2 • The closed linear manifold in 12 generated by all the rows of P is denoted by < P >. It is shown [7J that any intermediate closed linear manifold T has the form 00
00
,
00,
o },
(2)
where P is a m x
mxl
00
Hilbert matrix, and
00
Hilbert matrix such that
(3)
where P is a m x
289
290
S.]. LEI'.
(4)
<
-
PC
-1
>
= 9,Z 9
<
P
>.
The following shows that a natural pairing restricted to Tl (!) T; is decomposed by the linear functions defining any given intermediate closed linear manifold and the ones defining its adjoint. This is fundamental to a later development. THEOREM 1.
(Boundary-Form Formula)
Let P and P be m x
00
and ~ x
00
normalized Hilbert matrices satisfying (4).
Then
* *-1 - I -1 *-1 + * iB\a)P (PC C P*) PC (B (b))
for all a = {a 1 ,a) E T1>
b
* = {bz,b l } E To'
Atiyah and Singer [1] gave a general formula for the index of an elliptic operator in the case when the dimensions of null spaces involved are finite. It is expressed by a Chern characteristic and a Todd class. In the following we give a general formula for the index of a linear manifold, which is expressed by the dimensions of linear manifolds, This different formula is motivated from ordinary differential operator (see [8] also). THEOREM 2.
(Index Formula)
Assume that Null To and Null Tl* are zero dimensional and (5)
9,2
= {B(a)C + B+ (b)!a E Null Tl III {O}, bE Null To* III {O}}.
Take any closed linear manifold T with To eTc Il and write it as (2) for some m x normalized Hilbert matrix P. Put 00
m
= dim
9,z
9 < P >.
Then dim Null T + dim Null T: and
dimNullT * +m-
[NDEX AND NONHOMOGEN1:0US CONDrJ'IONS FOR MANIFOLDS
dim Null T* + dim Null T1
291
dim Null T + m.
m
Let us denote by ~2 the Hilbert space of 1 x m complex matrices a with aa* < When m = the simpler notation for ~~ is ~2'
~.
00,
THEOREM 3.
(Nonhomogeneous Boundary Va 1ue Problem)
Let P be a m x normalized Hilbert matrix and g ~ Range T1 , y we have the following: 00
(I).
(Necessity)
~~ be given. Then
If there exists an element x in the domain of Tl such that ~
{x,g} then
2
__
T1 , P(B({x,g})) *
_
bz(g)
=
*
iy (PP)
=
*
+
-1
yt ,
PC(B ({bz,O})) ,
for all {b 2 ,O} ~ T: which satisfy one of the following three equivalent conditions: (i)
where T
(i i )
{a
~
T1
I
P(B{a)) *
+ * B({bz'O})C
o }. mx l
~.
o
( iii)
00
(II) (Sufficiency).
x 1
Assume that (5) holds and that {B(a)P*la ~ Null T1 ffi {a}}
is closed in
m
~2'
Then the converse of (I) remains valid. We remark here that by subdividing an interval into infinitely many intervals, the above theorem is applicable to study the deficiency index of an ordinary differential operator. It is hoped that the theorem will find its application to partial differential operator and linear control theory. Proof of Theorem 1. (i) First we prove that if E is a x nonsingular Hilbert matrix, and if A and F m x normalized Hilbert matrices such that = < F>, then ~EF* is a m x m nonsingular Hilbert matrix. 00
00
00
292
S.]. LEE
(ii) Let Q and Qbe the m x --* QQ = 1m. Then
00
and m x
00
Hilbert matrices such that QQ* = 1m '
(iii) The right side of (3) is written as
*
~*--....
(iv) Express Q Q and Q Q as required in the theorem. Proof of Theorem 2.
(i) Let n
~
* dim Null T1, n* = dim Null To.
Let {cjJj' ... ,cjJn} b; a Besselian-Hilbertian basis for Null T1 , and let {\)Jj , ... ,\)In*} be one for Null To. Let G be the n x ro Hilbert matrix whose jth row is B({cjJj'O}). Let -G be the n* x Hilbert matrix whose jth row is B+({\)Jj'O}). Define two oper+ ators UT* and VT by 00
Then u~* and VT define one-to-one operators into Null (.PG*) and Null (.PG*) respectively. Here
(ii) The condition (5) implies that u~* and V are onto Null (.PG*) and Null (.pi;*) T respectively. (iii) Since VT and u~* are isomorphisms onto, it follows that
--*> dim Null T + dim
=
- , m
dim Null T* + dim = m. However, n = dim Null T + dim < PG *
>,
n* = dim Null T* + dim < PG*
>.
INDEX AND NONHOMOGI'NEOLTS CONDITIONS FOR MANIFOLDS
293
Combining these with the above, we get the results. Proof of Theorem 3. The necessity follows immediately from Theorem 1. We now prove the suffi ci ency. Let a1 E Domain T1 such that {a1,g} E T1 . Then for all (aj) E )!,n , we have a + { ZUj¢j ,O} E T1. Thus the converse is valid if 2
y - B({a1,g})
E
* Range (.GP).
Using the assumptions, this is equivalent to
for all b
{b 2 ,O}
E
* Null T.
However, this is always true by Theorem 1.
References 1.
M.F. Atiyah and I.M. Singer, The index of elliptic operators on compact manifolds, Bull Amer. Math. Soc., 69 (3)(1963),422-433.
2.
E.A. Coddington and A. Dijksma, Adjoint subspaces in Banach spaces, with appl ication to ordinary differential subspaces, Anal. Mat. Pura Appl. (4) 118(1978), 1 - 118.
3.
R.G. Cooke, Infinite Matrices and Sequence spaces, Mad1illan (1950), London.
4.
I.C. Gohberg and M.G. Krein, The basic properties on defect numbers, rootnumbers and indices of linear operators, Uspehi Mat. Nauk (N.S.) 12(1957), no. 2(74),43-118; Amer. Math. Soc. Transl. (series 2) 13(1960), 185-264.
5.
S.J. Lee, Operators generated by countably many differential operators, J. Diff. Equations, 29(1978), 452-466.
6.
S.J. Lee, Coordinatized adjoint subs paces in Hilbert spaces, with application to ordinary differential operators, Proc. London Math. Soc. 3(41)(1980), 138 - 160.
7.
S.J. Lee, Boundary Conditions for linear Manifolds, I, J. Math. Anal. Appl. 73(2} (1980), 366 - 380.
8.
S.J. Lee, Nonhomogeneous boundary value problems for linear manifolds and applications (to appear).
9.
I. Singer, Bases in Banach spaces I, Springer-Verlag (1970), New York.
This page intentionally left blank
This Page Intentionally Left Blank
Spectral Theory of Differential Operators I.W. Knowles and R. T. Lewis leds.) © North-Holland Publishing Company, 1981
ON THE POSITIVE SPECTRUM OF SCHRODINGER OPERATORS WITH LONG RANGE POTENTIALS Howard A. Levine Department of Mathematics Iowa State University Ames, Iowa 50011
We are concerned with solutions of the equation 6u + p(x)u = 0 in an unbounded domain Q c Rn which contains the exterior of a sphere. The principal theorem gives sufficient ~onditions on p(x) to ensure the nonexistence of nontrivial LL (at infinity) solutions. The principal application is to the determination of upper bounds for positive eigenvalues of Schrodinger operators. Our results include a result of Kato and of Agmon as a special case. Furthermore our potentials need not be radially symmetric.
In this paper we shall discuss some recent results obtained jointly with G. B. Khosrovshahi and L. E. Payne on the positive spectrum of Schrodinger operators with long range potentials. The details are contained in our joint paper [4 ].
I. NOTATION. Let Q C Rn , n > 3 be an unbounded open, connected doma in containing the exterior of a sphere, {x E Rn I I xii = (x.x. /" > R). Let V be a real 1 1 0 valued function on Q and let H be any self adjoint realization of -6 + V where n 6 denotes the n dimensional Laplacian. From time to time we will also let n r = 11";(11. II.
THE RESULTS OF KATO AND AGMON.
I r V (x) I
(a)
<
Kato [2J showed that if
K for
r -> R 0
then the equation Hu had no nontrivial L2('l) solution if
Ie >
= AU
K2.
On the other hand, Agmon ([1 J, Theorem 4) showed the following for V E L~oC(rl) : If (i ).
V is locally Holder continuous in a connected open set has measure zero and Qo :J {x I I xii > Ro} and
Q
o
C ~,
Q -
Q
0
(ii). For r .:: Ro ' V(x) Vo(x) + Vl (x) where Vo(x) is continuous and has a continuous radial derivative such that 295
296
HOWARD A. LEVINE
(r -)- +00) (b) (c)
lim sup r dV lor = ~ r -)- +00 0 0 1 V1 (x) = 0 (r - - d (r -+ +00,
some
then H has no eigenvalues where A >
€
> 0),
~o/Z.
The principal result we set forth here includes both of these results as special cases. III. THE RESULTS OF KHOSROVSHAHI, LEVINE AND PAYNE. the following result (Theorem 4 of [4J).
In [4J we established
Suppose V E L1oc(Q) and is real valued. Assume that V satisfies (i) of Agmon's result and that for some R0 sufficiently large and all x, /lx/l > R , - 0 (iil'. V(x) = Vo(x) + V (x) + VZ(x) where Vo(x) is real, continuous, l possesses a continuous radial derivative and (a), (b) above hold, V (x) is real l and satisfies (a) above, and VZ(x) is such that (writing ~ = r~) p
(d)
sup f oV(o~) do ~ M < % for all Iltll=l r
p,
r ~ Ro .
Let a =
max
I I
{K + [K 2 +
2~o(1-ZM)2ii}Z
2K2 +
~0(1-4M)
L
4(1 - ZM)Z 2(1 - 4M)Z 2 Then H has no L (Q) eigenvunction corresponding to A if A > a.
l J
Notice that if V = Vo(K = M = 0) we recover Agmon's result, while if Vl (110 = M = 0) we obtain Kato's result. Observe also that condition (d) can hold for potentials Vz which are not solely functions of the radial variable. V
IV.
THE
Let u E
~lAIN
THEOREM.
In [4J we establ ished the following result.
2,00(Q) n L2(Q) be ~ solution of W f1 u
n
where p(x)
E
=0
L1oc 2 (Q), ~ real valued and can be written, for Ilxll
for some sufficiently large R*. (A)
+ p(x) u
sup Ilxll>R*
Irpl(x)I~K
Suppose that
>
R* > R0as -
297
POSITIVE SPECTRUM OF SCHROHDlNGHR OPERATORS
(8)
sup
Iltll=l
If
p
ap2(a~)dal ~ M < ~,
p,
r .:: R*
r
(C)
po (x) is areal with a continuous radial - continuous function -- - derivative such that -po (x) -> Kl > O.
(D)
r apolar + (2 -
(E)
y -
4M - [1 )po(x) - K2/~
<
£ for some
y >
O.
2 r aPolar + 2(1 - 4M - £3)Po(x) - 2K /(1 - 4M - £4) .:: £5 where the £i's tend to zero ~ R* + +00. Then.!i pix) i2- Holder continuous on ~ connected open subset of r, r 0 say, and meas (,' - [Jo) = 0, it follows that u = O.
If we identify pix) with A - V(x) so that po(x) = A - Vo(x), Pl (x) = -Vl (x), P2(x) = -V 2 (x), then as r + +00, po(x) ~ A, r aPolar + - Ao. Conditions (O,E) then yield (at r = 00) (D' )
- 1\
(E' )
- 1\
o o
+ A (2 - y +
4M) - K2 /y
>
0,
2>-(1 - 4M) - 2K 2/(1 - 4M)
>
o.
The conclusion of Section III then follows after choosing the optimal
y
in
(0' ).
In order to establish the Main Theorem we make use of the following Lemma which has appeared in different forms in various places [1,3J. Lemma I. Let F(t) be ~ nonnegative function on (O,toJ, continuous there and twice continuously differentiable on (O,t ). Let c ' a , a , £ be constants with o l l 2 c l > 0, £ > 0, a2 > 1 and al + a2 > 1 + 2c l . If (*)
t
fOn
o
-a2
F(n)dn
and (**)
F(t)F"(t) - (F' (t))2 .:: - clF(t)F' (t)/t + d-
al
[F(t)r l /n-
a2
o
F(n) dn
then F(t) - 0 on [O,t o ). In order to apply Lemma I, we let u be an L2 solution of and set F(t) f i p-2(n-l) u2 ds dp r S p
6
n
u
+
p(x)u
0
298
HarVARD A. LEVINE
where t = r-(n-2) and where S denotes the(n-l)-sphereof radius p. We need to p establish (**) only for to sufficiently small (R* sufficiently large.) (The 2 establishment of (*) is not hard if u E L (Q). To facilitate the computation we note that since u E L2(Q) and satisfies (1.1), 1 im inf r g; Pou2 ds = 0 (Al) r 7 '" Sr 2 0 (A2) 1im i nf r ~ \grad u\ ds r -)-Sr 00
1 im inf r g; r 7 '" \ Direct computation then yields ('
u ur ds
(A3)
0
d/dt)
FF" - (F,)2 .::. 2(n - 2)-2 F(t)
J'" ~ r
(\grad u \2 -
Sp
where up = aU/dp, The integral on the right hand side can be shown to have a lower bound of the form required by the right hand side of (**) provided one makes use of (A-E), employs the integral inequalities (81-C3) of the Appendix and the following lemmas. 2 Lemma II. Let U solve IInu + p(x)u = 0.ill Q 2..ill!. be .ill L (Q). If. P satisfies (A-E), then for R* sufficiently large, and ~ r > R*,
where 2C = y + n - 2 (1 - M) + E for some y > 0 and where small ~ one pleases if. R* h sufficiently large.
E
>
0 ~ be made ~
Lemma II may be established by means of a Rellich type identity as follows: R
J
r
~ (pu p + CU)(lIu + pu) ds dp = O. Sp
An integration by parts followed by the use of the inequalities 81, 82, C1, C2, C3 of the Appendix then gives the Lemma. Finally one employs Lemma III.
R
Let u, p be
~i!!
Lemma II.
Then
'"
J 1 J P \grad u\2 ds do dp
r
p
p
S0
~lz(l+d
J r-
S
o
12 u ds + (1+E)
J -1 J P (po+du 00
r
00
p p
S
0
2
ds do dp.
299
POSI1'lVE SPECTRUM Of' SCHIWEDINGLR OPl'RArORS
These are combined to yield the desired lower bound for FF" - (F,)2 (see[4J). The computations are very tedious. V.
SOME EXAMPLES AND REMARKS.
Example 1.
Let V(r)
Then if we take V(r)
=
Ar~ sin(r B).
V2 (r), condition (d) of Section III becomes IA JP 06+1 sin(00)dol 2
~A r 2+6- S
r
So if 2 + 6 < S, we may take M = (2A/S)(R*)2+6-S as small as we please if R* is sufficiently large. Thus -6 + V has no eigenvalues \ E (0,00). Example 2.
Let V(r)
where A, B are positive constants. von Neumann and Wigner.
=
~r sin (Br)
This includes the class of potentials of
We find that if we take
(i)
V
V2 ' then there are no eigenvalues in (0,00) if 2A/B
(ii)
V
Vl ' there are no eigenvalues in (A 2 ,oo) if y. ~ 2A/B
(iii)
V
Yo' there are no eigenvalues in (izAB,oo) if ZA/B .:: l.
<
y.; 1;
The nature of the spectrum in (O,a) is unknown in general. l The above example shows that a depends upon how the potential is decomposed. The result of Section III has its analog in one space dimension when Vz = O. That is, if M = 0 the result is exactly the same as that of Eastham [6]. On the other hand it is not known that if (b) is replaced by the stronger condition (b)'
lim sup r laV larl = r
->-
+00
0
A 0
then the improved result of Knowles [7J for the one dimensional case, namely Z d + V(x)
-dl
has no eigenvalues in (a,oo) where 4
a = !zfKZ + (K
holds in the three dimensional case as well.
+ A;)iz),
300
HOW1RD A. LEVINE
APPENDIX. The following integral inequalities were established in [3, 4] for solutions of 6 nU + p(x)u = O. They use the conditions on the pieces P ' P2 of l the potential given in (A), (8). The constant E is generic but decreases with R*. Also R > r > R* > Ro where R* is taken large. R
\ J ¢ r
Sp
R 2 PPPluudsdp)
f
R
P
Rl
R
r
r
f ¢u YrS
R
So
p
K2
+
R
2
f ¢ p2u r
S
dsdp!
M <
R
p
¢
S R
R1
Y r
2
R
(83)
ds
€
R
p p
2 2 ~ [u + u ] dsdp, S P
J
r
f 1 f ¢ (2MPo S
u dsdodp
(84)
a
R
+
2
~
S
P p
(el)
p
~
r
dsdp,
2
"2 f - f
u
R
2
(u o ) dsdcrdp
SR +
(81)
p
n
r
R
K2
Rl
So
p p
u ds dp,
Sp
dSdp+"2
if - f p
~
2
f ¢
E
p
f P oPluuodsdodpl
J -
R
~
2 P1 u ds dpl
I UU R ) ds
2 + s)u dsdodp
a
R
R
f 1f ~
+ (2M + s)
r
p p
Igrad ul 2 dsdodp
(C2)
So
for computable constant C, R
I J ~ ppzuu dsdpl 2 M ~ luu \ ds r
S
p
P
S
i
+
R
R
+
f P (MP O +
r
s)u 2 dsdp
Sp
R
+
ds
P
(M + €)
f P Igrad r
Sp
ul 2 dsdp,
(C3)
301
POSITIVE SPECTRUM OP SCHROt;WNGER OPliRATORS
R1 R If - f f 0P2 UU dsdodpl r p S 0 p 0
<
-
R C 1n (~ ) cj u2ds + M 1n ( r) cj Iu I luRlds r S SR R R R 2 + f 1 f P (MPO + du ds do dp r P p Sa + (M +
R1 R
2
d f - f P Igrad uj ds do dp r
P p
-
S 0
(C4)
for computable constant C. REFERENCES [lJ
Agmon, S., Lower bounds for solutions of Schrodinger equations, J. Analyse Math. 23(1970) 1-25.
[2J
Kato, T., Growth properties of the reduced wave equation with variable coefficients. Comm. Pure Appl. Math 12(1959) 403-425.
[3J
Khosrovshahi, G. B., Nonexistence of nontrivial solutions of Schrodinger type systems, SIAM J. Math Anal. 8(1977) 998-1013.
[4J
Khosrovshahi, G. B., Levine, H. A., and Payne, L. E., On the positive spectrum of Schrodinger operators with long range potentials, Trans. Am. Math. Soc. 253(1979) 211-228.
[5J
Simon, B., On positive eigenvalues of one body SChrodinger operators, Comm. Pure Appl. Math. 22(1969) 531-538.
[6J
Eastham, M. S. P., On the absence of square integrable solutions of the Sturm-Liouville equation, Proc. Conf. Ordinary and Partial Differential Equations, Dundee, 1976. Springer Verlag Lect. Notes Math. 564(1976) 72-77.
[7J
Knowles, I., On the location of eigenvalues of second order linear differential operators. Proc. Roy. Soc. Edin. 80A(1978) 15-22.
It is known that there is at most one eigenvalue in this interval for the Wigner-von Neumann potential. The author thanks Professor A. Devinatz for this information.
This page intentionally left blank
This Page Intentionally Left Blank
Spectral Theorv of Differential Operators I.W. Knowles and R. T. Lewis (eds.) © North·Holiand Publishing Company, 1981
THE SPECTRA OF SOME SINGULAR ELLIPTIC OPERATORS OF SECOND ORDER Roger T. Lewis * Department of Mathematics University of Alabama in Birmingham Birmingham, Alabama 35294
The Friedrichs extension of a second order singular elliptic operator is considered on a weighted L~(Sl) space. We consider both the case in which the region Sl is bounded and the case with Sl unbounded. Necessary conditions and sufficient conditions on the coefficients that will insure a discrete spectrum are given with a certain degree of sharpness achieved. The boundary conditions include the Dirichlet, Neumann, and mixed Dirich1etNeumann boundary value problems. 1.
I NTRODUCT ION
Let h be a symmetric closed form bounded from below by a positive number with a domain D(h) that is dense in a Hilbert space H. The first representation theorem [9 , p. 322] guarantees the exi stence of a se1 fadjoint operator T/1 associated with It with the same bound from below. Our primary concern in this paper is with conditions that will insure that Th has a discrete spectrum, i.e., Th has a compact resolvent and the spectrum consists of a countable number of eigenvalues of finite multiplicity which converge to~. We refer the reader to the paper by John Baxley in these proceedings for related results using different methods. Let Sl be an open, connected subset of lR n that is not necessari ly bounded. Denote the boundar.y of Sl by r. Let w(x) be a measurable weight function that is positive-valued for almost every x E u. The Hilbert space of complex-valued functions f(x) satisfying f w(x) If(x)1 2 dx < Sl will be denoted by L~(Sl). 00
We shall have occasion to write ,l as the union of an increasing sequence of open sets {Slk}' Slk ::: Slk+l for each k. Let Hh(Sl) and Hh(Slk) denote the innerproduct spaces formed from the elements of D(h) and {¢ISl : <j> E D(h)}, respectively, k with the form inner product (u,v)h" Ir[u,v]. Lemma 1. Suppose that Sl is the union of an incr'easing sequence of open sets {[lk} fO]o which the identity injection \ : Hh([lk) -7 L~(Slk) is compact. If theY'e is a ['ositive-valued function p(x) on [l and a sequence of positive number's 303
304
Sk
ROGER T. LEWIS
~
0 as k
~
00
such that
w(x) p(xf
1
<
sk fop almost every
X E
~ 'V ~k
(1)
and
J
p(x) lu(x) 12 dx
<
h[u,u] for all u
E
(2)
D(h)
"'V~k
then Th has a discrete
Lemma 1 nally due to Glazman [5, it is stated
spectI~m.
is known in many different special cases. It appears to be origiFriedrichs. See, for example, Dunford and Schwartz [4, p. 1448], pp. 89, 152], and Schechter [16, p. 188]. A proof for the lemma as here can be found in [11].
The Sobolev space Hm(u) is defined to be the completion of = {fl_: f E C~(Rn)} with respect to the norm ~.~ m induced by the inner u H (~) product I {j Oaf Dag: lal ~ m}. (f,g) m
Cm(~)
H (0)
(l
The space H~(~) is the closure of C~(u) in Hm(~). The next two hypotheses will be assumed in much of what follows. The reader is referred to Showalter [18] or Treves [19] for explanation of some of the terms. Hl:
Assume that u is the union of an increasing sequence of bounded open sets {~k} each of which lies on one side of its boundary, f k. For some positive integer m, assume that {¢Iu : ¢ E D(h)} ~ Hm(u k ) k
for k = 1,2,···. Finally, assume that each fk is a Cm-manifold and that w(x) is bounded on each ~k. The next hypothesis is a coercivity requirement. H2:
For each k there is a c k > 0 such that 2 Ih[u,u]1 ~ c k ~u~ m
H (~k)
for all u
E
{¢I" : ¢
E D(h)}.
k
Corollary to Lemma 1. below by a positive number.
Let;., be a closable, symmetric form bounded fpom Let;., have closure h, and assume Hl and
H2.
If
(1) of Lemma 1 holds and
p(x) Iv(x)1 2 dx ~ ;.,[v,v] for all v
J
u'Vu
E
(3)
0(;"),
k
then Th has a discpete spectpWTI.
Proof. First, we show that (3) implies (2) of Lemma 1. If u sequence {v n } E D(~) such that vn ~ u in L~(~) and
there~
E
D(h) then
305
SPECTRA OF SOME SfNGULAR ELLfPTIC OPERATORS OF SECOND ORDER
Z - V , V - V J + 0 for n, m + Therefore, {v } has a limit in L (" m n m n p and by (1), that limit is also in L~(Sl ~ "k)' Therefore, ul,,'01l is the
s[v
00.
n
L~(n
"k) - 1 imit of {v }. Inequal ity (Z) now follows since n ~[vn' vnJ + h[u,uJ as n + =. '0
~
"k)'
k
The identity injection from Hh("k) to Hm("k) is continuous for each k by H2. By Hl, Hm("k) can be compactly imbedded in L~("k) - see Showalter [18, p. 49J. Hence, the identity injection \ :
Hh(~lk)
L~("k) is compact.
+
By Lemma 1,
the proof is complete.
A well-known application of this corollary is the one in which the potential q(x) becomes unbounded at = on an unbounded set n ~ Rn with sufficiently smooth boundary, which implies that the spectrum of the Schrodinger operator -6 + q(x) n
It is clear from the above corollary that a
has a discrete spectrum [5 ,16J.
similar application holds for the higher order Schrodinger operator (_l)k 6 k + q(x). n We assume throughout the remainder of this paper that the boundary of ", r, is sufficiently smooth in order that the first formula of Green applies. note the trace operator [18, p. 40J on the Sobolev space Hm(n) by y
o
We de-
and the
traces of the normal derivatives by Yj' j = 1,2,···,m=1. Let 1·1 denote the Euclidean norm in lR n (as well as absolute value - the difference being clear from the context). The next lemma, whose origin dates to the Hardy inequality, will be used to establish condition (3). In the case of n = 1, the proof is established in [7 J for compact support functions q,(x). 2
Lemma 2.
f "
for aU ¢(x)
Let 9 E H (,,) be real-Dalued and satisfy 6 g(x)
n
16 ng(x)1 1q,(x)1 2 dx:: 2
f n ::,~ f
.
E
C~(lRn) that satisfy (-1)11
f r
0 on ", then
1q,(x)1 Il7g(x)1 117
Ylg(s) Yo lcp(s)1
2
ds
<
0
ng > 0 on 11 and 11 = 1 if "'n 9 < 0 on ". Proof. By the first formula of Green, f6ng(~X) 12dx = frl 9(S)YojCP(s) 12ds - 2 fl7g(x)' (Recp(x)I7'¢\xT)dx " r 11 ~ 2 ~ll7g(x II Icp(x) I Il7cp(x) Idx
IJhere
11
=0
s,
t-
if
6
~ 2[f16 n 9(x)1 Icp(x)12dxi'[fll7g(x)IZI6ng(x)I-1117
"
by the Cauchy-Schwarz inequal ity.
"
The conclusion now follows.
(4)
(5)
306
ROGER T. LEII'IS
The lemma obviously holds for all ~ E C~(Q), which will later allow us to apply it to the case of the Dirichlet problem. However, for some sets Q and certain choices of functions g(x), a much wider range of applications is possible. Since ylg = '7g.", where" denotes the unit outward normal at xEr, then '7g'" ~ 0, for each x E r, will insure that inequality (5) holds for all = n ~ E Co(lR ). Example 1. Suppose that, for some a E Cl(Rl) and r = lxi, g(x) = air) for all x E r. Assume that either r does not contain the origin or that a' (0) = O. Then, the inequality a' (r)(x l ,'" ,x n ) . ,,~ 0, X E r, (6) implies that (4) and (5) hold for all ~ E C~(lRn) provided "'ng > 0 on Q. Example 1 follows from the fact that Ylg any nonzero x E r.
=
I7g'"
=
r-la'(r)(xl,···,x )·" for n
In some of our appl ications, when g is a radial function as above, a' (r) will be nonnegative. In order to insure that inequality (5) holds for all ~ E C=(lRn) we will need to have r = r- u r O where r- and r O are defined as o follows: . + r O, an d r Defl-ne r,
to be the set of all x E r such that
(xl" .. ,x n )· ("1'" . '''n) is positive, zero, or negative, respectively, where" is the unit outward normal vector at
= ("1 ,'"
'''n)
x E r.
Note that x is in r+, r O , or r- according to whether the angle 8 between the point vector from the origin to x and the outward normal vector" is acute, right, or obtuse. For example, if Q is the exterior of a ball that is centered at the origin, then e = ~ and r = r . Corollary to Lemma 2. 4
J
If
~
E
C1 (n '" {O}), then o
Ixl S 1'7~(x)12 dx.:: (s - 2 + n)2
J
IxI S-
2
1~(x)12 dx.
n Q Mar'eover, for n > 2, inequality (7) is valid for all 1 n + ~ E {u E Co (lR '" {O}): u (x) = 0 on r } when B > 2 - n and when S
valid for aU
~
E
{u E
C~(lRn
<
(7)
2 - n, it is
'" {o}): u(x) = 0 on r-}.
Proof. The proof is trivial for S = 2 - n. If sis - 2 + n) f 0 let s and apply Lemma 2. If S = 0, and n f 2, let g(x) = n ~ 2 lnlxl. g(x) remainder of the proof follows from the discussion above.
-=-IxI
Inequality (7) is proved in [3 J for ~ E C~(Q '" {On and in that paper an earlier proof is attributed to Piepenbrink [14J.
The
307
SPECTRA OF SOME SINGL'LI R ELUP'l'IC OPliRA FORS OF S};'COSD ORDER
2.
SUFFICIENT CONDITIONS FOR DISCRETENESS OF THE SPECTRUM OF SECOND ORDER ELLIPTIC DIFFERENTIAL OPERATORS.
In this section we assume that the boundary of Q, f, is a Cl-manifold and that n is the union of an increasing sequence of bounded open sets {Qk}' l '\ ~ [lk+l' each of which lies on one side of its boundary, f k , which is a C _ manifold. We do not assume that [l is necessarily bounded. Define the differential operator S by S = -w(x)
-1
{
n
d
i ,j=l
xi
L -a-
d
a .. (x)-3- + q(x)} lJ Xj
where A(x) = (a .. (x)) is a symmetric n x n matrix whose elements are continuously . lJ differentiable on n. The weight w(x) is defined as above (see Hl also), and q(x) is assumed to be a real-valued, measurable, locally integrable function on [l that has a positive lower bound on [l. (The requirement that the lower bound be positive can be relaxed some by adding A > 0 to the form ~ below.) Let 0(S) be a piecewise smooth, nonnegative function on f. Set n
d
n
t
d
_u_= L (L a .. (x)v.)_u_= (l7u) A·v dV A i=l j=l lJ J dX j where v = (vl ,'" ,v ) denotes the unit outward normal on f. Finally, we define n the domain of S to be ro n 2 D(S) = {u: u = ¢In for some ¢ E Co(lR), Su E Lw(Q), the support of u is in [lk for some k, and
c(s)~ + 0(S)U(S) = 0 for s "VA
E f}
where c(s) is equal to either 1 or 0 for each s E f. Assume that c(s) and 0(S) are not both zero at any s E f. The differentiable operator S: L2 (Q) ~ L2 (Q) is closable since its domain, w w which contains C~(Q), is dense in L~('i) [9, p. 268]. As a consequence, the form ~ defined by ~[u,v] = (Su,v) with 0(6) = D(S) is closable [9, p. 318] and by Green's formula ~[u,v] = f[(A(x)l7u(x),l7v(x)) + q(x)u(x)v(x)]dx + j0(s)u(s)v(s)ds. [l f Let h be the closure of
1~ng 1
and the 1imit
1I7g 12
~
., . 1 ue A() mlnlmum elgenva x,
X E Q,
(8)
308
ROGER T. LEWIS
Firstly, we consider the case in which Q is unbounded. Let ~A be a positivevalued function on [0,00) satisfying minimum eigenvalue A(x) 2- ~A(lxl), x E Q. ~he eigenvalues of A(x) can be regarded as continuous functions on Q - see [ 9, Ch. II- §5. 5J.) Suppose that there is a sequence of positive numbers {c k}, (possibly converging to zero), such that ~A(lxl»_ck>O'
XEQk'
Note that h satisfies Hl and H2. When n = 1, it is shown in [2] (see [8], [10], and [5, p.120] also for w(x) _ 1), where the higher order case ia also considered, that for Q = [1,=) and q = 0, Th has a discrete spectrum if, and only if 1im W(x) 1 X-koo x
or
~A(t)
1
where
dt
0
=
I
when
'"
1
x
1
lim W(x) X-koo
-1
~A(t rl dt = 0
f
when 00
W(x)
dt
when
wet) dt
when
1
1
1
-1
~A
wet) dt
dt
< '"
dt = 00 <
00
00
x
1
f
'"
-1 ~A
1 1
wet) dt = 00.
This result is extended to the case when n > 1 in the theorems that follow. For the sake of simplicity we assume that p = inf{ Ixl: X E Q} > S > 0 for some number S. Theorem 1.
Let Q be unbounded.
Assume that either r+ =
0 or
that u(s)
0
Let w(r) be a positive-valued, piecewise continuous
on r+ for all u E D(S).
function on (0,00) satisfying
w(x):"w(r),
XEQ,
r= Ixl.
(9)
Assume that there is a number K > 0 such that for all X E Q 00
1 sl-n
~A(s)-ldS
r
r
1 sn-l
5 co w(s)[J t n- 1 w(t)dt f t l - n ~A(t)-ldt]-l
ds <
K. (10)
s
S
p
If r
lim
1
r-- S
t n- l w(t)dt
J
r
t l - n ~A(t)-l dt = 0
(11 )
then Th has a discrete spectrum.
Proof.
Let g(x)
X E
11,
and hex)
lxlrl-n/rsn-lw(s)[/Stn-lw(t)dtJOOtl-n~A(t)-ldt]-ldSdr, p
p
S
s
X E Q,
309
SPECTRA OF SOMH SINGULAR ELLIPTIC OPERATORS OF SECOND ORDER
then Ivgl 2
Ixl lin g
IVhl2
Ix1 2- 2n [
f=
Ix1 2- 2n [
sl-n ~ (s)-l ds]-2, A
= \l A( Ix In)
-1
2
Ivg I ,
Ix I s f sn-lw(s)(f tn-lw(t)dt
li h = w( n Hence, for any u E O(s)
=l n t - \lA(t)-ldt)-lds]2,
s
S
p
and
f
IXI~IJltn-1W(t)dt f=tl-n~A(t)-ldt. Ixl
S
S[u , u] ::
f ~ A (I x I)
Ivu 12 dx
(l
:: f
Ivg 12 ( LIng) -
1
IVu 12 dx
(l
.:: 2-
1
f
Ivg I IVu I Iu I dx
(l
.:: (2Kr
l
f
IVhl Ivul lui dx
(l
.:: (4Kr by Lemma 2 and inequality (10). the Corollary to Lemma 1.
l
f
(l
LI h lul n
2
dx
The conclusion now follows from (9), (11), and
The rather annoying requirement posed by (10) is not very restrictive when viewed in conjunction with (11). (The author conjectures that the requirement (10) can be eliminated.) For example, if the product of (11) decreases to zero then it is easy to show that (10) is satisfied. More generally, it is not hard to show that (10) is satisfied provided there is a function F(r) such that for any
X E (l S
=
[f tn-lw(t)dt f tl-n\lA(t)-l dt]-l S
~ /(Ixl) + [Ijltn-lw(t)dt s
and
(tl-n\lA(trldtrl,
Ix I
Ix I 1 f sn- w(s)ds
s
<
Ix I,
=
s l-n ( r l ds Ixl \lAS is bounded for all x E (l. If F(r) is a constant function then this requirement specifies that, as the inverse of the product in (11), r = [f t n- l w(t)dt f t l - n ~A(t)-l dt]-l, S r diverges to infinity, the negative oscillation remains bounded. F(lxl)
p
f
The next theorem shows that for certain weight functions an analogous sufficient condition for a discrete spectrum holds even when
310
ROGER T. LEWIS
p
Theo rem 2. on r- foy' aU u
Let E
Q
be unbounded.
f If n
=0
Assume that ei ther r
01'
that u (s )
0
Let inequa~itu (9) hold and
D(s).
sn-l w(s)ds
< "'.
(J
2 assume that
f
fr
(r
p
If
s
-1
)JA(s)
-1
dS)
-1
dr
< "'.
(J
Ijlsl-nlJA(SrldS /'" sn-lw(s) ( (tn-lw(t)dt / tl-n)J(trldtrlds P Ix I s (J
(12)
is bounded on G and
1 im r-)-oo
f
r
'" t n-l
l n w(t)dt fr t - lJA (t) -1 dt 0
0
(13)
then Tfl h.as a disCf'cte spectrum.
Proof.
Let '" 1 r l-n -1 -1 g(x) = f r -n [f s )lArs) ds] dr, Ixl (J
h(x) = _;xlsl-n f"'tn-lw(t)( f"'un-lw(u)dU p s t
ft
ul - n lJA(u)-ldu)-l dtds,
(J
and then proceed as in the proof of Theorem 1. It should be noted that when )JA(t) = t 2- n and w(t) = t B for B < -n the limit in (13) holds but the function in (12) is not bounded on \1. However, it is shown below that Tfl has a discrete spectrum even in this case. Theorems 1 and 2 obviously remain valid if (11) and (13) are replaced by the more general conditions
d
and
lim {w(xrlq(x) + Ix 1->-'" i3
xl
tn-lw(t)dt
lim {w(xrlq(x) + [f tn-lw(t)dt Ix 1-+ Ix I respectively. In the special case of \1 = R n and w(x) 00
(tl-nlJA(trl dtrl) = '" Ixl Ix I
f tl-nlJA(trl dtrl}= "',
(14)
(15)
(J
= 1, a theorem of Schechter [16, p. 192] (ef. Lemma 3.3 of [17J) can be used (see [llJ) to remove condition (10) of Theorem 1. Consequently, we can conclude from the Corollary to Theorem 4 below that when Q = JRn, q(x) = 0, w(x) a 1, and min e.v. A(x) = max. e.v. A(x)= )lA(lxl), then
311
SPECTRA OF SOM}, SINGULA /( ELLIPTIC OPhRA TORS OF SECO.'VD ORDER
lim Ix 1-+
Ixl n (l-n ()-l dr Ixl r iJA r
00
=
0
is necessary and sufficient in order that the spectrum of Th be discrete. Secondly, we illustrate the method for the case in which Q is bounded and the singularities of s occur on a portion of r. Assume that Q = Qn-l x G where n l Qn-l ~ R - and G C (O,p] for some p > 1. Assume that the singularities of S occur onl y on r n {x E Rn : x = O}. (We refer the reader to the book of n t~ikhl in [12, p. 207] where M. M. Smirnov has considered a similar problem.) Let w be a function of one variable such that w(x) ~ w(x n ) for x E Q. Let iJ A be a nonnegative function of a single variable satisfying min e.v. A(x) -> iJA(X n ) for x E Q. Suppose that for each k there is a ck' such that iJA(x n ) ,::c k > 0 for x E Qk. Note that the case in which iJA(O) = 0 is included. We would need only to choose each Qk in order that fk does not intersect the plane xn = O. For example, this problem appears to be associated with the study of heat flow along a rod with an end (x n = 0) which is completely insulated (see Mikhlin [12, p. 156]). The case in which iJA(x n ) -+ as xn -+ 0 is also included. 00
Theorem 3. Let
f
Assume that u(s)
x
n iJjil(S)dS
p
ff
w(t) (
xn
o \7..
f
1
t
= 0 on
r- u {x E r: X n t 1
f
w(x) dx l ·· ·dx n
Qn- 1
= O} for aZl
IJ; (s)ds)-
1
U E
O(S).
dt be bounded on
a
If
-1
t
p
lim f f w(x) dxl···dx n f IJ A (s)ds t-+O+ t Qn-l a
0
then Th has a discrete spectrum. 1 t -1 )-1 Proof. Let g(x) = f (f IJ A (s)ds dt and
xn
1
h(x)
=
f f
1
0
p
w(s) (
xn t
ff s Qn-l
w(x) dx l ·· ·dx n
f
s
_1 -1 IJ A (v)dv) ds dt.
a
The proof now
follows the proof of Theorem 1. Theorem 4. Let
f
p
Assume that u(s)
xn
1
iJji (s)ds J
xn bounded on
Q.
=0
on f
+
{x
E f:
xn = o} for all u
( ) dx 1 ·· .dx n f IJA-1 (s)ds )-1 dt (f f x w() tw a Qn- 1 t
If
t
w(x) dx 1 ·· .dx n
o Qn- 1 then Th has a discrete spectrum.
Let g(x) =
f
1
xn
E
O(S).
p
f f Proof.
U
t
(f t
p
1
iJji (s)dsr
1
f
p
1
iJA (s)ds
t
dt and
o
(s
>
0) be
ROGER T. LEWIS
312
s
t
J
h(x)
w(s)
s
3.
( JJ 0
Qn-l
p
w(x) dx l " 'dx n
J
-1 ~A
(v ) dv )-1 ds dt.
s
NECESSARY CONDITIONS FOR DISCRETENESS OF THE SPECTRUM OF SECOND ORDER ELLIPTIC DIFFERENTIAL OPERATORS
In this section, we show that the theorems of the last section are sharp, at least in certain cases. Our main device for doing this will be the following theorem that can be found in the book of Glazman [5 , p. 15]. Theorem 5.
A necessary and sufficient condition for the nwnbe" of points of
A, lying to the left of a given point Ao'
the spectrwn of a self-adjoint operator
to be an infinite set, is that theroe exists an infinite dimensional set G C D(A) for which
(Au - Aou, u) < 0 for all U E G. Since -6 of section 2 is symmetric then This self-adjoint [9, p. 323] and Theorem 5 applies to Th. Corollary to Theorem 5.
If there is an infinite dimensional set M C D(I1)
such that
h(u, u)
<
Ao(u, u)
j"or aU
U E
M,
then the nwnber oj" points of the spectr'wn of Th' lying to the lej"t oj" Ao' is inj"inite.
The proof follows by letting G = D(T I1 ) n M - see [llJ. Lemma 3. Let U be a bounded open subset oj" Q that lies on one side oj" its l boundary, which is a C manifold. Suppose ~ E Co(U) has a piecewise continuous derivative in U. Ij" q and ware bounded on compact subsets of Q, then ~ E D(h). Proof.
Let
¢ be
jump discontinuities, then which implies that
~
as zero outside U. Since ~ has at most aX. 2 1 1 n L (:R n ), for i = 1,2,··· ,no Hence, ~ E H (:R )
the extension of
=
;Iu
¢. aX. ~¢
H~(~)
E
E
~
(see Treves [19, pp. 245-247]).
Consequently,
~ is the Hl(U)-limit of C~(U) functions {~k};=l' Since q and ware bounded on supp ~ and each a ij E Cl(Q), then {~k} is h-convergent to ~ [9, p. 313] which implies that
~ E
D(h) by Theorem 1.17 of [ 9, p. 315].
In order to prove necessary conditions for the discreteness of the spectrum of Th analogous to the sufficient conditions of Theorems 1 and 2, the following hypothesis will be required concerning Q. H3.
Let n {xE:R : Ixl'::S}~Q
for some number S
>
1.
313
SPECTRA OF SOME SINC['LAR ELUl'TIC OPERATORS OF SECOND ORDER
Theorem 6. XE{XElR
Let YA(r), ;;;(r) , q(rl
Assume H3. n
: Ixl::S}
E
C[S,oo) and assume that {Oy,
maximwn eigenvabe A(x) ~ YA(lxl), w(x)~;;;(lxl),
q(x)~q(lxl)·
If they'e is a sequence {
of
continuous piecewise diffeY'entiable functions
lJith disjoint supports in [S,oo) such that
n l 2 ( r - [YA(r)l¢k(rlI + (q(r) -
Ao~(r»)I¢k(r)12Jdr
0
<
(16)
B
for
k, then
each
spectY'um (T I) II (-00, A ) rl
0
is infinite.
Proof.
The proof follows from the Corollary to Theorem 5, Lemma 3, the
i nequa 1 ity h [¢k ' cjlk J -
1.0
~ f [y A( I x I ) 1cl>k ( I xl) 12 + (q( I x I ) - AO~ ( 1xl»)
(cI>k '
2
I q, k ( 1xl) 1 Jdx
rl
for each k, and a change of variable to polar coordinates. It is interesting to note that for YA(r) E C[8,00), the sequence {
= r n - l Ao;;'(r)y(r)
(17)
is oscillatory [5 J (i.e., every solution has an infinite number of zeros on [8,00»
or, equivalently, if each self-adjoint differential operator generated by
(17) has an infinite number of points of its spectrum in the interval (-oo,A ) o (The continuity requirements on the coefficients probably can
see [1 , p. 525J. be relaxed.)
It is known (c.f. Ahlbrandt, Hinton, Lewis [2J, Moore [13J, Hille [6 J) that equation (17) is oscillatory provided q(r) -< A0 ;;'(r) on [8,00) and ei ther
f
r
n-l
(Ao;:'(r) - q(r»)dr
6
r lim sup f r+= S
s
l-n
and
< =
YA(s)
-1
00
ds
fr
s n- 1 ( AOW- ( s ) - q(s))ds
(18)
1
>
or
f
sl-n YA(s,-l ds
S
<
00 and
00
1 im sup f r-+oo r
s l-n YA(s) -1 ds
fr
s
n-l
B
(A 0 ~(s) - q(s»)ds
>
1.
(19)
Also, equation (17) is oscillatory [13J when
f
S
s
l-n
()-l YA s ds
= foo s n-l B
q(s»)ds
(20)
314
ROGER T. LErVIS
Consequently, corresponding to the sufficient condition for discreteness of the spectrum of Th given in Theorem 1 we have the necessary conditions of the next corollary. Corollary to Theorem 4.
Assume H3, q(r) _ 0 on
Suppose that Th has a discY'ete spectY'um.
u, and that y'A(r)
E
C[B,oo).
If
oo rn-l w(r)dr J
00,
<
B
then
If s J001-n
YA ()-l s ds
<
00,
B
then
lim
fr
-() s n-l w s ds Joo s l-n YA(s) -1 ds = O.
r-+oo B
r
Proof. Since the spectrum of Th is discrete then neither (18) nor (19) hold for any Ao > 0, i.e., either -( s ) ds ~ Ao-1 lim sup fr s l-n YA ()-l s ds foo sn-l w r-+oo B r limoo sup r-+
f
s 1- n YA(s) - 1 ds
r
Jr
- ( s ) ds s n- 1 w
~
Ao- 1
S
for every Ao > 0, depending on which integral exists. sion follows.
Consequently, the concl u-
In order to illustrate the sharpness of Theorems 1 and 2 and the Corollary to Theorem 6, we state the following example([see [11] for a proof). (Xl (X2 _ Example 2. Let ~A(r) = r 'YA(r) = r ,w(r) = w(r) = rK, and q(x) = Ixl T = rT for r = Ixl. (i)
(ii)
Sufficient Conditions: If (X1 > 2 - n assume that u = 0 on r+, if (Xl = 2 - n assume that u = 0 on r, and if (Xl < 2 - n assume that u on r for all u E 0(5). The spectrum of Th is discrete if (Xl > K + 2 or , > K. Necessary Conditions: Assume H3. then cx2 > K + 2 or , > K.
If the spectrum of Th is discrete
Therefore, in the case of the Dirichlet problem - c(s) r - and K n cx S = -Ixl- { I --"-- Ixl --"-- + Ixl'} i=l aXi "Xi with hypothesis H3 holding.
=0
and o(s) B 1 on
The spectrum of Th is discrete if, and only if,
0
315
SPECTRA OF SOME SINGL'LIn I,LLIPHC (JPToRATORS OF S};COND ORD1:R
a
> K
+ 2 or
T > K.
Next, we illustrate the above procedure for obtaining necessary conditions corresponding to Theorems 3 and 4. Assume that Q is bounded and that " = (x ERn: 0 < x < 1, - 1/2:: Xl':: 3/2, i = l, .. ·,n-l} is a proper subset of o n n Assume that q(x) := 0 on and that there is a function of one variable YA such that maximum eigenvalue of A(x)
"0
-
n
0
Our first theorem allows the weight function to become degenerate on xn = O. Assume that YA l~S Jcc;Y'eam:ng on (0,1). If the spectrum of Til is
Theorem 7. discrete then
1
t 1 1 1 lim J Y; (s)ds J J t+O+ 0 t 0
J w(x) dx l ' "dx n = O. o
Proof. We give the proof for n = 2. For n > 2, the proof is completely analogous. Suppose that the 1imit is not zero. Let B represent a constant that we will specify below. Then, for Ao chosen large enough there is a sequence tk -)- 0 such that tk 1 1 1 1 im J YA (s)ds J J k-+oo 0 tk 0 By Theorem 5 and Lemma 3 in order to show that the spectrum of Til is not discrete, it will suffice to show that for every 0 > 0 there are numbers a and d such that o < a < d < 0 and a function E (0([-1/2, 3/2J x [a,dJ) with piecewise continuous derivatives such that 1
J
3/2
f
o -liZ
[YA(x 2 )! IJ !
2
-
2
W(Xl'XZ)!! J dX l dX Z < O. a
A
In fact, this will show that the spectrum of Til has an infinite number of points in the interval (-00,\ ). o
Let 2(t + liZ) a(t)
{
-l/Z::t
o<
Z(~/2 - t)
<
t < 1
t
<
3/Z
a
<
t
<
b
b
<
c
and
f
a
t
b l YA (s)ds ( J YA-1 (s ) ds a
r1
s(t) d -1 J YA (s)ds t Define q,(xl,x Z) other x E Q.
d
J
c
YA-1 (s)ds rl
a(xl )S(x Z) for (Xl ,x 2 ) E [-1/2, 3/2J
c x
<
t -< d.
[a,dJ and
<j>
= 0 for all
316
ROGER T. LEWIS
Given 6 > 0, choose d E (0,6) and c = d/2. We will choose b = tK for some K. Calculations show that as b = tK ... 0 there is a constant B such that 1 3/2
f f
o -1/2 For some
E >
2
YA(x 2 ) 1'1q,1
dx l dx 2
b
_1
(f YA (s)ds)
<
-1
a
B.
0
f
lim k-+«> Consequently, as
tk
-1
0
b ...
1
YA (s)ds f f tk 0
Ao w(x l ,x 2 ) dx l dx 2 = B +
E.
0
1 3/2
2
f f
[YA(x 2 ) 1'11>1
o -1/2
-
b
(f YA1(s)dSr l
<
(21 )
a tK so that
Choose b
tK -1 1 1 YA (s)ds f f Ao w(x l ,x 2 ) dX l dX 2 o tK 0
f
>
B + E/2
<
E/8.
and tK
1
f
YA (s)ds f
o
Choose a
< b
1
c
1
f Ao w(x l ,x 2 ) dX l dX 2 0
so that all
1
f YA (s)ds f f Ao w(x l ,x 2 ) dX l dX 2 tK
o
<
E/8.
0
Hence, the right side of inequality (21) is negative and the proof is complete. Theorem 8.
Assume that YA is increasing on (0,1) and that 1 1
f ." f
o
w(x) dxl· .. dx n
0
<
00
If the spectrum of Th is discrete then
1
_1
t
lim f YA (s)ds f f t-+O+ t o o Proof. n
Define
a,
1
f
o
1
w(x)dxl· .. dxn=O.
S, and q, as in Theorem 7.
Calculations show that (for
= 2)
By choosing c = tk for some k, the proof will follow in a manner similar to the proof of Theorem 7. The monotonicity requirements on YA of Theorems 7 and 8 can probably be weakened as in Theorems 5 and 6. Theorem 8 shows that the spectrum of Th is not discrete if w(x) a 1 and YA(x n ) = x~ for a ~ 2. This special case is due to a
317
SP£CTRil OF SOME SiNGULAR JiLLlPTlC OP£RATORS OF SECOND ORDER
result of Mikhlin [12, p. 2llJ. REFERENCES [1]
Akhiezer, N. I. and Glazman, I. M., Theory of Linear Operators in Hilbert Space Volume II,(Pitman Advanced Publishing Program, Boston, London, Melbourne, 1981.)
[2J
Ahlbrandt, Calvin D., Hinton, Don B., Lewis, Roger T., Necessary and sufficient conditions for the discreteness of the spectrum of certain singular differential operators, Canadian J. Math, in press.
[3]
Allegretto, W., Nonoscillation theory of elliptic equations of order 2n, Pacific J. Math. (1976), 64, 1-16.
[4 J Dunford, N. and Schwartz, J., Linear Operators Part II,(Interscience Pub-
lishers, Inc., New York, 1957.)
[5]
Glazman, 1. M., Direct ~lethods of Qual itative Spectral Analysis of Singular Differential Operators, (Israel Program for Scientific Translations, Jerusalem, 1965.)
[6]
Hille, Einar, Nonoscillation theorems, Trans. American Math. Soc. (1948), 64, 234-252.
[7]
Hinton, Don B. and Lewis, Roger T., Discrete spectra criteria for singular differential operators with middle terms, Math. Proc. Cambridge Phil. Soc. (1975), 77, 337-347.
[8]
Kalyabin, G. A., A necessary and sufficient condition for the spectrum of a homogeneous operation to be discrete in the matrix case, Differential Equations (1973), 9, 951-954. (Translation of Differensial'nye Uraneniya).
[9]
Kato, T., Perturbation Theory for Linear Operators, Second Edition (SpringerVerlag, Berlin, Heidelberg, New York, 1976).
[lOJ Lewis, Roger T., The discreteness of the spectrum of self-adjoint, even orde~ one-term, differential operators, Proc. American Math. Soc. 42 (1974) 480-482. [llJ Lewis, Roger T., Singular elliptic operators of second order with purely discrete spectra, submitted for publication. [12J Mikhlin, S. G., Linear Equations of Mathematical Physics (Holt, Rinehart and ~Jinston, Inc., New York, 1967). ' [13J Moore, Richard A., The behavior of solutions of a linear differential equation of second order, Pacific J. Math. 5 (1955), 125-145. [14J Piepenbrink, J., Integral inequalities and theorems of Liouville type, J. Math. Analysis and Applications, 26 (1969) 630-639. MR 39, #3136. [15J Rel1ich, Franz, Perturbation Theory of Eigenvalue Problems (Gordon and Breach Science Publishers, New York, London, Paris, 1969). [16J Schechter, Martin, Spectra of Partial Differential Operators (North-Holland, Amsterdam, London, 1971). [17J Schechter, Martin, On the spectra of singular elliptic operators, 23 (1976) 107-115.
Mathematik~
318
[18J
ROGHR T. LEWIS
Showalter, R. E., Hilbert Space Methods for Partial Differential Equations (Pitman, London, San Francisco, Melbourne, 1977).
[19J Treves, Francois, Basic Linear Partial Differential Equations (Academic Press, New York, San Francisco, London, 1975).
* The author was partially supported by NSF grant number MCS-8005811.
Spectral Theory of Differential Operators I.W. Knowles and R. T. Lewis (eds.) © North·Holland Publishing Company, 1981
RECAPTURING SOLUTIONS OF AN ELLIPTIC PARTIAL DIFFERENTIAL EQUATION Peter A. McCoy Department of Mathematics United States Naval Academy Annapolis, Maryland 21402
2 2 2 Let k -C(x +y )~ 0 b~ regular on tbe closure of the disk 2 D:x +y2 < I and in D let ¢ be a regular solution of the 2 2 Schriidinger's equation {'J2+(k - C(x +y2)}¢ = O. When the
restriction of ¢ to the
~D
is sufficiently smooth, expan-
sion formulae are given that recapture and shifted means on an arc of the
aD.
~
from its means A solution is
given that interpolates prescribed means at the boundary.
INTRODUCTION In a series of papers, C. H. Ching and C. K. Chui [2-5] develop mean boundary value formulae and uniqueness theorems for recovery of select analytic functions of a complex-variable that are defined on a disk, an annulus and certain conformal equivalents. Some of these characterizations were extended to axially symmetric potentials in En (n ~ 3) by Peter McCoy [10,11] with the aid of the Bergman-Whittaker integral operator [1]. And then, extended to a class of axially symmetric elliptic equations in E3 by R. P. Gilbert's Method of Ascent [8].
The topic considered directly concerns the recovery of select regular 2n-oeriodic solutions of the Schrodinger equation L(~) = {3 2/dr 2 + l/r "i)/dr + 1/r2 32/'Je 2 + (k2_C(r2))} <jl = 0 (1) (in polar coordinates (r, e)) from data averaged at sets of equally spaced points distributed along the boundary of a disk, an arc of the boundary and the boundaries of certain conformally equivalent domains. The basis for the analysis is the extension of the characterization of analytic functions found in C. H. Ching and C. K. Chui [3,5]. PRELIMI NARI ES Let us limit the coefficient k2_C(r2) to be real-valued, non-positive and regular on the closure of the disk D:r < 1. S. Bergman [1] and R. P. Gilbert [7,8] define the complete set of functions
o
(2) 319
320
PETER McCOY
2- n j+1E(r 2 ,t) (1_t 2 )n-l/2 dt , n ~ 0
J * (r) n
-1
in terms of which regular solutions of eqn. (1) expand as uniformly convergent * are analytic and J * (1) , 0, n > 0, a conseseries on compacta of O. The In(r) n quence of the regularity of C(r2) [see 1]. Let the class R be the linear space of (single-valued) regular solutions ¢ in 0 with continuous extension to the cl(D). Properties of class R are drawn from the linear space A of (single-valued) analytic functions in 0 with continuous extension to the cl(D). Let the class B (E > 0) designate those continuous 2nl E periodic functions g on dO whose Fo~rier coefficients satisfy an = O(l/n + ) where in8 , B E [0,2n) . g(e iB ) L an e n=O For f E AE
AilB E, Ching and Chui [2,5] define the basis l/Jo(z)
(3)
l/Jn(z)
=
jJ(%)zs , n
L
1,
-
1,2, ...
=
sin in terms of which the Riemann series expansion of f, f(z)
anl/Jn(Z)' zED
L
n=O
converges uniformly and is the restriction of f to the dO. Using this information as the starting point we define the functions (4)
n= TC '
L.
sin
jJ(~) ¢ (r,B) , L(~ )
s
s
n
=0
These may be derived from the ~n by the linear map TC (l/Jn)' that is defined by the Hadamard products
1, . . . . ~n =
¢
S
(r, e) = P (z)
~n(r,B)
a
* ZS
s = 0,1,2, ...
= Po(z) * l/Jn(z) , n = 0,1,2, ...
taken with ZS , l/J n and the analytic Poisson kernel Po(z) = on (o,z) E 02 . (5)
n~o ~n(o)zn
,
The natural map TC:f
~n(reiB) +
= ¢n(r,8) , n > 0 ,
¢, f E A,
.e
¢(r,e) = Pr(z) * f(z) , z = re 1
,¢(r,o)
1:
n=O
a
~
n n
(r,e)
is clearly one-one and by Hopf's maximum principle, it is uniformly convergent on compacta of D. Hopf's maximum principle also assures that 2 Re Po(z) ~ c>O, (o,z) EO. Therefore, by Korovkin's theorem [9]on positive
321
RECAPTURING SOL( TIONS OF .4N ELLIPTIC EQl'A'FJON
operators,the eqn. (5) extends continuously to the dO since the associate f does. The map TC sends the class A one-one into the class R. Let f sA, then by the above ep = TC(f) E R = rUiB (<:>0) because s ie i8 ep(l,e) = fee ), was established by the iden~ity ~ (~,e) = ~ (e ), n > O. More.e n n over, if ep E Rand feel ) = ep(l,o) , then the results of [5] apply to cons truc t E a (unique) f E A for which 1> TC(f). Indeed, the linear spaces A and Bare E E E isomorphic under the map TC' Conti nui ng . in thi s di recti on, we defi ne the sup-norm I I9 II r = sup {lg(pe1G)I:(l = TC(f), E 1 lIepllr.". IIfll ' r.:: 1. But when 1> E RE and f = TC- (ep), II f ll r .:: Ilepll ' r < 1 so that 111>11 = Ilfll. To summarize, Theorem 1. The linear space ArlB E of analytic functions and the linear space Rn BE of regular solutions of L(ep) = 0 are isometrically isomorphic for each fixed E > O. THE MEAN BOUNDARY VALUES Approximate solutions with error bounds are constructed from smooth data at equally spaced points on the aD. The construction extends to data at points along a subarc of the aD at the loss of the el'ror estimates. Conformal equivalents and an interpolation problem are investigated. The constructions focus on the arithmetic means n
0n(g;8 1 ,8 2 ) = lin
L
k=l
g(exp (i21Tk(8 Z-8 1 lin + i21T8 1 ))
and
n=1,2, ... of a continuous function g on an arc {e 21Tis ; e .::s.::s 2}. l 0n(g;O,l) ~n(g;O,l), n = 1,2, ... , o,,,(g;o,1) = lim 0n(g;O,l).
Note tha t The shifted
n->=
means of g, . n-l \!n(g;O,o) = g(e 21TOl )/2n + lin L g(exp(i21T(2k-l)6/2n-l), k=l 21Ti8 n=1,2, ... app 1y to a proper subarc {e ;0'::S.::6}, 0<6<1, of the aD. The mean boundary value problem is solved next. Theorem 2. (6)
Let the function ep E
R~
~(r,8)
=
BE (c>O).
I
n=O
p
n
Then the Riemann series expansion
(~)~ (r,e) n
PETER McCOY
322
(7)
p
n (q,) = C5 n (q,;0,1) -
C5 co
(q,;0,1), n>O,
p
0
(¢) =
C5
, 00
represents q, uniformly on the cl (D). If C5 n(¢;O,l) = 0 for all n Furthermore, the following estimates are uniformly valid in e k 0 I¢ (r, 8) - 2: p (rjJ) 'JI (r, 8 )1< K( 6, iJl ) k(8) n=O n n for all k
~
1, 6 <
E
for all k
~
0, then q, - O.
and r = 1, and liJl(r,e) -
(9 )
~
k E
n=O
p
n
(¢)'JI (r,B~< K(¢)k- E (1_r 2 )-1 n -
1 and r < 1.
Proof. Let f = TC -1 (q,) E AE be the TC - associate of ¢. Since each Riemann coefficient Pn(q,) = Pn(f), f is represented [5] by the uniformly convergent Riemann series expansion f(z)
Z Pn(¢)wn(z) , zEcl(D) .
n=O
However, f(e i8 ) = ¢(1,8) so that as noted earlier from Hopf's principle, eqn. (5) is valid. Again from the maximum principle the appraisal k
1¢(r,8) = If(e
k
P {¢)'JI (r,e)I
ie
) -
k
E
n=O
p
n
(f)w (e n
i
e
P
n
{¢)'JI (l,e)1 n
)1 = Mk(f,e), k=O,l, ... 6
Application of the bounds in [5] verify that Mk(f,e) 2 K(o,¢)k- for all -E 2-1 k ~ 1, 6 < E and r 2 1; and that by [9] Mk,r(f,G) 2 K( ¢ ) k (l-r) for all k > 1 and r < 1.
Having established the basic representation theorem, we direct our attention to the interpolation problem. This is another consequence of the extension of the series representation of 'C(f) to the dD. Theorem 3. Let {a } and {B } be sequences of real numbers that converge to a n n 3+ 2+E) and 0 respectively, with the rates an-a = O(l/n E) and Bn = O(l/n for some E > O. Then there exists a unique function iJl E R + E ,for some E' > 0 such that: 2 (i) L(¢) = 0 in D and
for all n=l ,2, ... (10)
Furthermore, the series n=l (an-a)An(r,e) + n=l Bnrn(r,e) + a
converge uniformly to q,(r,e) on the cl (D).
Here, the basis is
RliCAPTURlNG SOH'nONS OF AN liLLlPllC EQUA nON
(~n(r,e)
+
(~n(r,e)
~n(r,e)}
323
/2
'n(r,e)} /2i
for all n=1,2, .... Proof.
From the sequences {an} and {Sn}' construct the functions h(z) =
(11 )
(a
L
n=O
-a)~
n
n
(z), k(z) =
L
n=O
S ~ (z) . n n
These are analytic [see 5] so the representations H(r,e) = Te(h(z)), K(r,e) = Te(k(z)) are uniformly convergent in the cl (0) and L(H) = L(K) = 0 in O. Because d2/1e2~ (r,e)=d2/a82~ (r,_e)=a2/a82~ (r,e),s>O , ss s the conjugate functions H = Te(h) and K = 'e(k) are solutions of L = 0 where 'e(~) refers to the expansions in eqn. (11) conjugated. Oefine the following functions: U(r,e)
=
[H(r,e) + H(r,e)]/2
V(r,e) = [K(r,e)
'e((h+h)/2)
K(r,e)]/2i
If 1> = U+iV = 'e(f) , f = (h+h)/2 + (k-k)/2i, then L(
= O. Also, f E A2+E , (E' > 0) by [5] and so tha t E A2+E , . Because of the growth of the coefficients, on r = 1 the expansions of ~ and f can be differentiated termwise iS8 ) , s = 0,1, ... , so that in e and a/ae s(l,e) ~ alae (e ie 1S 3/as (l,s) = a/38 f(e ). Recalling the identity ~(l ,e) = f(e ) on the boundary, we apply [5] to verify (ii). Turn our attention to those solutions that are characterized from means taken over an arc of the boundary. We see that a representation formula can be developed. Theorem 4. Let tha t ei ther
~ E
R
E
and
~e
be continuous on the arc {e
i
e :0~e~o},0<6<1, such
or n (1);0,6) = vn(~;O,o) = 0, is satisfied for all n=1,2, ..... Then ~ a 0 on the cl(D). (ii)
0
Proof. Previous reasoning concludes that if f = 'e-l(
324
PETER McCOY
(12 )
1: { 2m/ 2m- 1v (
m=l
m
- a (
m-
00
1 ( 1, e/ 6- 1)
which is a uniformly convergent expansion on the interval [-8,6]. The limit is the function A(e) = 1: b A (1 ,e/6-l). However, A is the (unique) restriction of n=Q n n -1 the TC - associate f on [0,6]. Apply the formula [3] to recover TC (
(13)
a 1J! (z) n n
1:
n=Q
,,->=
1
e i t+z e -z
dt
~
where A is the uniform limit. convergent Riemann series
Next, for each ,,>0 develop fEAE as the uniformly
f" (z) =
on compacta of O.
,,,>0
1
n=O
an "iJ!n(z) '
For z = re i8 , take the Hadamard product
(14)
=
l:
n=O
a
n,
,,'I' (z)
n
l. e
with = by the Hopf maximum principle, the fact that an,,,-+a n and because of uniform convergence in eqn. (15), limits interchange. The representation formula is then
l:
n=O
an
'
,,'I'
n
(r,e) } .
In closing, we examine conformal maps that extend the preceeding theory to other domains. Let W be an open (bounded) star-shaped set that is the conformally image of the open disk 0 under the map z = iJ!(w) and let the extension of iJ! to the dW be a twice continuously differentiable diffeomorphism. Such a set W is called a class S domain and is written as iJ!E S (w,O). Then the function g, g(w) = (foiJ!)(w) = f(iJ!(w)), ZEW, is analytic and extends continuously to dW for fER (E>l). Upon taking composition of the maps, the function E
00
!(r,e) = (
~O
* [iJ! o1J!](w)
plainly extends! and fOiJ! to the dW.
n
n=o an[iJ!n0iJ!J)(w) =
l:
~O
('I'
a n
n
oiJ!)(w)
The following result is direct.
RECAPTURING SUU "I lONS
or AN ELLIPTIC /'QUADOlV
Theorem 5. Let the function ¢ E R (E>l) and let E series expansion !(r,e)
( 15)
I:
=
n=O
p
~ E
S (w,D).
325
Then the Riemann
n(o~)('¥ no~)(r,8)
represents _¢ uniformly on the cl (w). If the 0 n (¢;O,l) = 0 for all n ?_ 0, then = O. Furthermore, the following estimates are uniform in 8
(16)
for all k (17)
k -6 n (¢o~)('¥ o~)(r,e)!
!!(r,o) >
1, 8 <
E
I:
and r < 1, and
!!(r,e) -
k
I:
n=O
p
n
(o~)('¥ o~)(r,e)
n
!
for all k > 1 and r < 6 , the minimum distance between the image of the set r=o -1 and the set r=l under the map ~ REFERENCES [1] Bergman, S., Integral Operators in the Theory of Linear Partial Differential Equations, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 23, Springer-Verlag, New York, Inc. 1969. [2]
Ching, C. H. and Chui, C. K., Uniqueness Theorems Determined by Function Values at the Roots of Unity, J. Approximation Theory 9 (1973) 267-271.
[3]
, Analytic Functions Characterized by Their Means on an Arc, Trans. Amer. Math. Soc. 184 (1973) 175-183.
[4]
, Recapturing a Holomorphic Function on an Annulus from its Mean Boundary Values, Proc. Amer. Math. Soc. 41 (1973) 120-126.
[5]
, Mean Boundary Value Problems and Riemann Series, J. Approximation Theory 19 (1974) 324-336.
[6]
Colton, D. L., Solution of Boundary Value Problems by the Method of Integral Operators, Research Notes in Math., vol. 6, Pitman Publishing, San Francisco, 1976.
[7]
Gilbert, R. P., Function Theoretic Methods in Partial Differential Equations, Math. in Science and Engineering, vol. 54, Academic Press, New York, 1969.
[8]
, Constructive Methods for Elliptic Equations, Lecture Notes in Math., vol. 365, Springer-Verlag, New York, 1974.
[9]
Lorentz, G. G., Approximation of Functions, Holt, Rinehart and Winston, New York, 1966.
[10]
McCoy, P. A., Mean Boundary Value Problems for a Class of Elliptic Equations in 0, Proc. Amer. Math. Soc. 76 (1979) 123-128.
[11]
, A Mean Boundary Value Problem for a Generalized Axisymmetric Potential on a Doubly Connected Region, Jour. Math. Analysis and Applications 76 (1980) 213-222.
This page intentionally left blank
This Page Intentionally Left Blank
Spectral Theory of Differential Operators I. W. Knowles and R. T. Lewis (eds.) © North·Hol/and Publishing Company, 1981
FOURTH ORDER INVERSE EIGENVALUE PROBLEMS Joyce R. McLaughlin Department of Mathematical Sciences Rensselaer Polytechnic Institute Troy, New York 12181 A well posed inverse eigenvalue problem is discussed here. It is assumed that two positive sequences Al < 1. '" and PI' P 2 , ... are given which have prescribed 2 asymptotic forms. Construction of unique coefficients l 3 A(s) EO C [O,l] , B(s) E" C [0,1] which depend continuously on the sequences Ai,Pi,i = l,2,3, ... , is given. The A. 's and p. 's are the eigenvalues and normalization l
l
constants for the problem y
(4)
+ (Ay
(1)
)
(1)
+ By - AY =
°
0, y(O) = y(l) (0) = 0, y(2) (0) = 1, y(l) = y(l) (1) = An integral relationship assumption between solutions of the derived problem and a known eigenvalue problem yields uniqueness. Interpretation of this relationship is given. INTRODUCTION Suppose the self-adjoint eigenvalue problem y(4) (1)
+ (Ay(l))
(1)
+ By _ AY
4
Z j=l
0,
°< s
< 1,
4
°, j=lZ
M.. y(j-l) (0)
lJ
N .. y(j-l) (1) = 0 , i = 1 , 2
lJ
is considered.
If A, B, M.. , N .. , i = 1,2,j = 1,2,3,4 are (real) given, then it lJ lJ of eigenvalues (which satisfies a 1 2 particular asymptotic form) can be determined. Suppose, however, that the inverse problem is considered. That is, suppose that a sequence of positive real numbers Al < 1.2 < •.• is given (satisfying a particular asymptotic form). The is well known
[5] that a sequence 1. ,1. ""
question then is whether or not there exist coefficients A and Band Mij,N i = 1,2, j = 1,2,3,4 so that the given sequences Al < 1.2 <
ij are eigenvalues
for (1). More particularly, a well posed problem is sought. That is criteria (or hypotheses) are sought in order to construct A,B,Mij,N , i = 1,2, j =1,2,3,4 ij uniquely and so that the resultant coefficients vary continuously with continuous changes in the sequence of Ai's. Before pursuing this further, it would be useful to review what is known for second order, self-adjoint, inverse eigenvalue problems. The second orderproblem has been studied much more extensively than the fourth order problem. In the second order case the self-adjoint eigenvalue problem to be considered is y(2) + (A - q)y (2)
2
0,
a <
s < 1
2
O:.y (j-l) (0) J
L: j=1
°, j=lL:
B.y(j-l) (1) J
= 0,
The sequence Al <
1.2 < ... satisfying certain asymptotic forms is given and the
coefficients q(s),
0: . ,
J
Bj ,
j = 1,2 are sought so that the given sequence of Ai'S
327
328
JOYCE McLAUGHLIN
contains all the eigenvalues for (2). Historically, extensive work on the problem has been done by Borg [4], Marcenko [15], Krein [9,10), Levinson [12), and Gel'fand and Levitan [6]. It is known that this problem is not well posed. The first example given by Borg [4] shows that knowledge of the eigenvalues is not enough to produce a unique set of coefficients. Uniqueness can be obtained if additional information is assumed. For example, one can assume knowledge of a second sequence of eigenvalues [4], [9], [10], [12], [15] where boundary conditions are different (but related) to those in (2). One could assume that q is symmetric about s =~, [4), [7], [8]. Or alternatively, unique solutions have been shown to exist, by Gel'fand and Levitan [6], when knowledge of a positive sequence P1,P2' ... ' which are shown to be normalization constants, is also assumed. The existence theorem obtained by a constructive process by Gel'fand and Levitan, [6], can also be applied to show existence when knowledge of two sequences of eigenvalues is assumed. This is done by applying the results of Levitan [13), [14) which state that given the two sequences of eigenvalues in [4], [9], [10], [12], [15], the normalization constants associated with either sequence of eigenvalues may be constructed. Finally, continuity results are much less complete. Barcilon [1] has obtained a continuity result when q is assumed to be symmetric about s = ~ and to have small L2 norm; Hald [7) and Borg [4] have obtained continuity results when q is symmetric and changes in the sequences of eigenvalues are sufficiently small. Work on the fourth order inverse eigenvalue problem has been done by, for example, BarcHon [2], [3), McKenna [16), Leibenzon [11], and the author [17), [18). Bard_Ion follows an approach of M. Krein assuming knowledge of three distirict sequences of eigenvalues and associated boundary conditions. Uniqueness is proved Also a constructive technique is given when it is known, a priori, that the given sequences are eigenvalues for eigenvalue problems which contain the given corresponding sets of boundary conditions. Leibenzon also proves a uniqueness theorem when three given sequences of eigenvalues and three given corresponding sequences of normalization constants are given for eigenvalue problems with related boundary conditions 4
The present paper discusses a completion of the problem of developing a well posed inverse eigenvalue problem utilizing the previous results of the author [17], [18]. The basic notion is a generalization of the work of Gel'fand and Levitan [6]. As a lead for the subsequent ideas, consider the following intuitive description of the Gel'fand-Levitan technique. Two sequences are given which satisfy required asymptotic forms. A known (base problem) eigenvalue problem is given. Then a linear integral relationship (with unknown kernp}, K) is shown to exist between solutions of the differential equation in the known eigenvalue problem and solutions of the differential equation in the (to be) derived eigenvalue problem. Using the given sequences,an integral equation for K is determined. Finally, the coefficients q and S., a., j = 1,2, are determined in terms of K; and the given J
J
sequences are the eigenvalues and normalization constants for the derived problem. It should be emphasized that it is not assumed a priori that the given sequences are eigenvalues and normalization constants for an eigenvalue problem. A solution for the fourth order inverse eigenvalue problem can be obtained by generalization of the successful second order Gel'fand-Levitan technique. The method of solution shall be illustrated by considering the following special problem. It is assumed that two positive sequences are given, i.e. Al < A2 < •.• and 3
P ,P , . . . . We seek coefficients A(s)€ C [0,l) and B(X)€Cl[O,l) such that the 2 l eigenvalue problem y(4) + (A/ l )) (1) + By _ Ay =
°
(3)
y(O)
y(l) (0)
y(l)
y(l) (1)
°
329
FOURTH ORDIiR INVERSE EIGENVALUE PROBLEMS
has eigenvalues Al < A2 < ... and the corresponding eigenfunctions (normalized by the condition y(2) (0) = 1) have the p. 's as normalization constants. ~
It can be
shown, unlike the second order case, (the example is given in section 1) that the two given sequences do not determine A and B uniquely.
An additional assumption is made, as follows.
Consider the known (base problem)
eigenvalue problem (4)
z(4) - AZ = 0, z(O) = z(l) (0)
z(1) = z(l) (1) = 0,
Look for coefficients A and B in (3) so that particular solutions of the differential equation in (4) are related by a linear integral relationship (with kernel K) to particular solutions of the differential equation in (3). The sequences Ai's and Pi's are used to determine K uniquely by a constructive, iterative technique. Then the coefficients A and B are determined in terms of derivatives ofK; and again the Ai's and Pi's are the eigenvalues and normalization constants for (3). The iterative teChnique produces a solution to some inverse problem at each iteration and in the limit produces a solution to the desired inverse problem. That this is true is shown in [17,18]. The results are reviewed in Section 1. It can also be shown that the coefficients A, B, and A(l) vary continuously with respect to continuous changes in the Ai's and Pi's provided the changes are small. In fact, this result can be expressed as a bound on the coefficients, that is a co
bound on the L
norm of the coefficients, where the bound is given in terms of the
£1 norm of the char,ges in the A. 's and the inverses l
~-
Pi
's.
This result is given
in Section 2 along with a brief explanation of the proof. Finally it would be useful to examine the integral relationship assumed between solutions of (3) and solutions of (4). It is shown in Section 4 that the assumption of this relationship implies a relationship between the eigenvalues and normalization constants for non-self adjoint problems related to (3) and (4). Section 1. In this section an iterative technique will be described for obtaining a solution of the fourth order inverse problem when sequences < Al < A2 < ... , and
°
Pi>O,i =1,2, ... are given.
The solution is unique when an additional "integral"
assumption is made. An example is discussed to show that for fixed sequences Ai'S and Pi'S a different solution to the inverse eigenvalue problem can be obtained when the "integral" assumption is changed. tion of the example, are contained in [17,18]. We begin with the known (base problem) problem (4). eigenvalues for this problem and
The results, with the excepLet Al* < A2* < ••• be the
* P * ,P2'
... the corresponding normalization conl 2 stants (that is, the square of the L norm of the eigenfunctions) when the eigenCunctions are normalized so that z (2) (0) = 1. Let zA (s) satisfy (5)
Z(4)
_ Az = 0, z(O)
1.
We will seek a sequence of eigenvalue problems (6)
y(4) + (Any(l)
(1) + Bny _ AY = 0, yeO) = yell (0) =
°
= yell
= yell (1),
n = 1,2, ... , such that the eigenvalues and normalization constants for the nth eigenvalue problem are Al < A2 < .. ,
*
*
< An < An + l < ... and P l ,P2,···,Pn ,Pn+l' P *+ , ... That is, the first n eigenvalues and normalization constants are from n 2 the given sequences and the remaining eigenvalues and normalization constants are
330
JOYCE McLAUGHLIN
the same as those in the base problem.
n
It is further required that solutions YA
of the differential equation in (6), which also satisfy yeO) = yell (0) = 0,
0, and y(2) (0) = 1, are related to the z\'s, for A > 0, by
yell
n
y~(s) = z\ (s) + JS K (s,t) z\ (t)dt
(7)
o
n where K is to be determined along with An and Bn. With these assumptions, no particular asymptotic forms for the sequences are needed to show the existence and uniqueness of An and Bn.
Particular asymptotic forms for the Ai's and Pi'S are
required in order for the An'S and Bn'S to converge to A and B, respectively, and so that the solutions, y\, of the differential equation (3), which satisfy y.(O) = y(l)(O) = yell = 0, y(2) (0) = 1, are related to the z\'s, for A > 0, by y\(s) = z\ (s) + J
(8)
s
K(s,t) z\(t)dt.
°
This last condition is again used for the uniqueness result for A and B in (3). We shall state the results described above. First for each n we let
f
(9)
n
(s,t)
n l: i=l
[ "\.l '*\.l 't'
zA~(s)z\~(t)
l
l
p.*
Pi
l
and let
(10)
f(s,t)
l: i=l
[ "\.l '""\.l 'C) ",,"'",,'" l l *
Pi
]
]
Pi
Then, Theorem 1:
Suppose that K (s,t) is continuous for O < t < s < l . n
Then
{y~ (s)}n
u{y~*(s)}w ,as defined in (7) is a complete orthogonal set on i=l i i=n+l * * < s < 1 with normalization constants P , . .. , Pn,Pn+l,Pn+2"" iff Kn (s,t) is the l unique solution of the integral equation i
°
f
(11)
(s,t) + JS Kn(s,u)f (t,u)du + Kn(s,t) = O. nOn
In addition, the resultant Kn(s,t) is analytic in sand t, 0 < t
* ,A +
< s < 1.
Also,
* * Pl"",Pn,Pn+l,Pn+2""
* '\n+2'···
\1'\2" .. ,A n n l and are the eigenvalues and n . normalization constants in (6) with corresponding eigenfunctions y\ ,l = 1, . .. ,n, n
.
y\*,l
i
n + l , n + 2 , ... , i f f
i
(12)
Remark:
3
n n d n - An(s) Kn(s,t) I +2(K _K ) -2 --3 K (s,s). s t=s ss tt t t=s ds The kernel in the integral equation (11) is degenerate.
Therefore,
Kn(s,t) tions.
can be determined simply by solving a set of linear, nonhomogeneous equaFinally, it can be shown that
(13)
Kn(s,t)
n l: i=l
[Y~~(S)ZA~(t) l
*
Pi
l
331
FOURTH ORDER INITRSIi UCENVALUE PROBLEMS
Theorem 2:
:, A~) '" + p, and 1
° < Al
Let
1
1 AiP
< A7 <
i
1
k
1
lk
L i=l
i=l
1.2 •... satisfy 171,)'"
+ R.1 with
* * A.P.
l:
and
1 ..
<
k = 0,1,2,3.4.
00,
Then there exists a unique solution K(s,t) (14)
f(s,t)
+ fS f(t,u)K(s,u)du + K(s,t)
e4 [0
€
< t < s < 1] of
°
=
°
with the property that (lj+k
aj +k
lim
--'-k-
n--
(ls] dt
j.k = 0,1,2,3,4,
°
~ j
--,-1<
Kn(S,t)
K(s,t)
dS] Clt
+ k < 4.
Further, YA.,i 2
=
1,2, ... , as defined in (8) •
1
forms a complete, orthogonal set in L [0.1] with normalization constants iff K(s,t)
satisfies (14).
The sequences 71 ,71 , •.• and P ,P , ... are the eigenvalues and normalization con1 2 1 2 stants for (3) with corresponding eigenfunctions Y ., i = 1,2, ... iff A(s) and A 1
B(S) are defined by A(s) =-4 ~ K(s s) ds "
(15)
B(s) Remark:
I
d
3
It can also be shown that
l:
[y"le),,,I')
i=l
P,*
00
(16)
I
-2 K(s,s). -A(S) K (s,t) + 2(K -K ) s ss tt t 3 t=s ds t=s
K(s;t)
1
1
YA. (s)zA. (t) 1
1
p.
1
1
]
We have stated the main results showing the iteration procedure, which provides existence and uniqueness. An example will now be described which will show that a change in the integral relationship (8) [or (7)] will again provide a w-,ique set of coefficients A(s), B(s) which are different from A(s) and B(s). Example:
711 <
712*
Suppose we seek an eigenvalue problem (2) which has eigenvalues
< .•• and normalization constants P ,P * ,P * , .•• Theorem 1 provides unique l 2 3
1 coefficients Al(S), Bl(s), and a unique K (S,t), when (8) (or (7)) is assumed to hold. (We are assuming P > 0 and 71* t- A ,A > 0). 1 1 1 1 Suppose now we change the integral condition (8) (or (7) as follows. Let zA be solution of z(4) - AZ = 0 which also satisfies z(o) = z(l) (0) = z(l) (1) = 0, z(2) (0) = 1. We then require that there exists Rl(s.t) £ that functions (17)
ZA (s)
+ fS Rl (s ,t)
ZA (t) d t
o
,
e[o
< t < s < 1] such
A >
°
satisfy a fourth order equation (18)
B\ _ Ay 0, 0 ~ s ~ 1, eigenvalues, Y ' Y*, i = 2,3 •... A A
y~4) + (Aly(l)) (1) +
=
and that 71 ,71 * ,71 * , ... are are eigenfunctions, 1 2 3 . . . * * 1 i . wlth normallzatlon constants, P ,P2 P3' ... for the elgenvalue problem consisting l of the above differential equation and boundary conditions,
332
JO YCli McLA UGHLIN
~ yell (0) ~ yell ~ yell (1) ~ a
yeo)
(19)
What has been done then is only to change the set of solutions of z(4) - AZ in the integral relationship used to define solutions of y(4) + (Aly(l»
i'?y - AY ~
~ a
(1) +
O.
l The same proofs, used to obtain Theorem 1 can be employed to determine Rl, A , and Bl uniquely.
a
< s < 1.
Furthermore, it is not true that Al
= Al
and Bl
The proof of this is by contradiction as follows.
= Al, Bl = Bl, for y, = YA y,* = Y'*' Al l' Al Ai
Al
(8) and (17)
a i
< s < 1, then Rl _ Kl for
a
2,3,4, ... ,
< s < 1.
a
< t
= Bl
for
If
< S < 1 and
The two integral relationships
imply that
a
+ fS K(s,t)
a
* i ~ 2,3, . . . . The theory of Volterra integral equations yields A , Ai' l This last equation is false and the desired contra(s) - ZA(S), a ~ s < 1.
for A
zA
diction is obtained, when A ~ A . l section 2: In this section we will present a continuity result for solutions of fourth order inverse eigenvalue problems. The result which will be presented shows that A and B vary continuously as the A.' sand P.' s vary continuously from the 's and P ~'s. 1
t.
~
1
1
More particularly a bound can be determined on the L'" norm of A, A(l), and B in
*
1
terms of the differences Ai - Ai and
Pi sufficiently small.
1
*
as long as the differences are
Pi
The theorem will be presented along with a brief explanation of the proof. Theorem 3:
* p., * i ~ 1,2, ... be defined as before. Let z;\, A.,
Let
1 1 1
AEC
[0,1], BlOC [0,1].
constants for (3).
Let Ai' Pi' i ~ 1,2, ... be eigenvalues and normalization
Let YA satisfy the differential equation of (3) for
°<
s < 1
and the conditions YA(O) ~ y(~) (0) ~ YA (1) ~ 0, y(~) (0) ~ 1. Suppose that 4 K(s,t) E c [0 < t < s < 1] is defined by (16), A and B are defined by (15), and that the derivatives of K Tn (15) can be obtained by termwise differentiation. Let
L
(17)
i~l
Let A
sup { l~i
Let M
o<
max O<s
A.1* A.1 -X-:-' *' 1 A.1
1
~' 1
1
Then there exists 00 and N(A) such that when
00'
(18)
M ~ N(A)o .
Remark 1:
It is possible to determine the actual numerical value of N(A).
Remark 2:
There is justification for assuming the existence of
the resultant inequality holds only for "small enough"
O.
0 , that is, that 0 One way to see this is
333
FOURTH ORDER INVERSE EIGENVALUE PROBLEMS
to observe that M becomes unbounded if say Al,A2,A3,A4,AS' all approach A . If we 6 * Pi * i = 6,7, ... then N(A)a is clearly bounded and (18) also assume Ai = Ai' Pi' cannot hold. A proof of this theorem can be given, similar to that in [7], using the theory of Voltera and Fredholm integral equations. We first rewrite K as
K(s,t) =
'" [(ZA.-ZA"!'V\. (1 l l l + zA* Y __ Ai Pi i Pi
Z\(Y>-Y<)J
=
i=l
Pi
Then let f(S,t,A) be the Green's function for the non-self-adjoint eigenvalue problem z(4) - AZ = 0, z(O) = z(l) (0) = z(2) (0) 3 4 Let Y(S,A) = {sinh(A~s) - sin(A~s) }/A / .
0 = z(l).
There then exists Q > 0 such that for A. ,A.* > QM K are determined by equations such as l 1
bounds for individual terms in
fl
o -B(~) (YA.-YA~)dt]dt, l
and Y . (s) = zA. (s) A 1 1 If either A: or Ai
:5..
+
f 0
1
f(s,t,\) [-(A(t)y . A l
(1)
)
(1)
1
- By . (t)] dt. A 1
Q M, bounds for the individual terms in K are determined by
equations such as
[(\)~
-
(Ai)"']Y(S,Aj') + (Aj'-\) fSy(s-t,Ai)y . (t)dt A
o
1
and (1) (t)) (1) - By . (t) ]dt. Ai Al Combining the resultant bounds yields (18). Y . (s)
A
= zA(s)
l
Section examine section inverse
l
+ fly(s-t,A.) [- (AY
o
l
3: We would now like to return to the integral assumption (8) (or (7)) and it more closely. As a reminder, we have already shown, in the example of 1, that a change in this assumption produces a different solution of the eigenvalue problem.
What we observe as a result of these assumptions can be described intuitively as follows. The assumption of the integral relation (8) implies that the spectral data for an associated non-self-adjoint eigenvalue problem is the same when the 4 1 1 f' .. dl. ferentlal equat10n 1S . e1 t h er Z (4), -AZ = 0 or Z ( ) + ( Az ( ) ) ( ) + Bz - ' AZ0 = • To be more specific, let us determine the solution of the inverse eigenvalue problem given by Theorems 1 and 2 of Section 1. Further, let P. ,~"!" i = 1,2, ... be l
l
the normalization constants and eigenvalues (with associated eigenfunctions zi (s)) for the eigenvalue problem (19)
z(4) - AZ = 0, z(O) = z(l) (0)
=
z(2) (0) = 0 = z(l).
z.
The eigenfunctions zi (s) are normalized so that (3) (0) = 1, the adjoint eigen2 functions are normalized so that (1) (0) = 0 alnci p. is the L inner product of l.,a l,a 1
z.
z.
334
Zi
JOYCE McLAUGHLIN
Z.l.a
and
(4)
Y
(20)
Then the eigenvalues and normalization constants for
+ (Ay(l»
(1) + By _ AY
= o.
y(O)
=
y(l) (0)
=
y(2) (0)
are also Ai' Pi' i = 1.2 •... where the associated eigenfunctions ized by y(3) (0) = 1, and the adjoint eigenfunctions 9(1) (0)
y.l,a
=
y(l)
=
O.
YA
are normali are normalized by
= 1.
Finally, it should be noted that. using the techniques developed by Leibenzon [12]. it can be shown that there is exactly one pair of coefficients A(s) € cl[o.ll and B(S) € C [O,lJ such that the eigenvalue problems (3) and (20) have eigenvalues and normalization constants Ai' Pi' i = 1.2 •... and
~., l
P.• l
i = 1.2 •... respectively.
Hence Theorems 1 and 2 produce this set of
unique coefficients. REFERENCES
[1]
V. Barcilon. Iterative Solution of the Inverse Sturm-Liouville Problem. J. Math. Phys .• 15 (1974), pp. 287-298.
[2]
V. Barcilon. on the solution of inverse eigenvalue problems of high orders. Geophys. J. R. Astr. Soc .• 39 (1974). pp. 143-154.
[3]
V. Barcilon. on the uniqueness of inverse eigenvalue problems. Ibid:. 38 (1974). pp. 287-298.
[4]
G. Borg. Eine Umkerung der Sturm-Liol1villeschen Eigenvertaufgabe. Acta. Math .• 78 (1946). pp. 1-96.
[5]
E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations. McGraw-Hill Book Co., New York, 1955.
[6]
I. M. Gel'fand and B. M. Levitan, On the Determination of a Differential Equation from its Spectral Function, Izv. Akad. Nauk SSSR Ser. Mat., 15 (1951), pp. 309-360; English transl., Amer. Math. Soc. Transl., 1 (1955), pp. 253-304.
[7]
o. H. Hald, The Inverse Sturm-Liouville Problem with Symmetric Potentials, Acta Math .• 141 (1978), pp. 263-291.
[8]
H. Hochstadt, The Inverse Sturm-Liouville Problem, Corom. Pure Appl. Math., 26 (1973), pp. 715-729.
[9]
M. G. Krein, On a Method of Effective Solution of a Inverse Boundary Problem, Dokl. Akad. Nauk SSSR. 94 (1954), pp. 987-990.
[10]
M. G. Krein. Solution of the Inverse Sturm-Lionville Problem, Ibid., 76 (1951), pp. 21-24.
[11]
Z. L. Leibenzon, The Inverse Problem of the Spectral Analysis of Ordinary Differential Operators of Higher Order, Trudy Moskov. Mat. Ob~~., 15 (1966) pp. 70-144; Trans. Moscow Math. Soc., 15 (1966) pp. 78-163.
[12]
N. Levinson, The Inverse sturm-Liouville Problem, Mat. Tidsskr. B., 25 (1949), pp. 25-30.
[13]
B. M. Levitan, Generalized Translation Operators and Some of Their Applications, Fizmatigz, Moscow, 1962; English trans. Israel Program for Scientific Translations, Jerusalem and Davey, New York, 1964.
[14]
B. M. Levitan, On the Determination of a Sturm-Liouville Equation by two Spectra, Izv. Akad., Nauk SSSR Ser. Mat., 38 (1964), pp. 63-78; Amer. Math. Soc. Transl., 68 (1968), pp. 1-20.
FOURTH URDU? INVf!RSE EIGENV,1LUE PROBLEMS
335
[15]
V. A. Marcenko, Concerning the Theory of a Differential Operator of the Second Order, Dakl. Akad. Nauk SSSR, 72 (1950), pp. 457-460.
[16]
J. 11cKenna, On the Lateral Vibration of Conical Bars, SIAM J. Appl. Math.,
[17]
J. R. McLaughlin, An Inverse Eigenvalue Problem of Order Four, SIAM J. Math. Anal., 7 (1976), pp. 646-661.
[18]
J. R. McLaughlin, An Inverse Eigenvalue Problem of Order Four - An Infinite Case, SIAM J. I·lath. Anal., 9 (1978), pp. 395-413.
21
(1971), pp. 265-278.
This page intentionally left blank
This Page Intentionally Left Blank
Spectral Theory of Differential Operators I. W. Knowles and R. T. Lewis leds.} © North·Holland Publishing Company, 1981
STURM THEORY IN n-SPACE Angelo B. Mingarelli Department of Mathematics University of Ottawa Ottawa; Ontario Canada Dedicated to Professor F.V. Atkinson on the occasion of his sixtyfifth birthday. Two conjectures are formulated regarding a form of the Sturm Comparison Theorem for a second order vector differential equation. These results are verified in particular cases and it is noted that their validity would lead to a form of Sturm's theorem for both self-adjoint and non self-adjoint equations. I NTRODU CTI ON In 1930 Marston Morse [4J formulated a version of the Sturm comparison and separation theorems which, when applied to the vector equation y" + Q(t]y
o
(1.1)
where Q(t) = Q*(t), Y E R n , yielded a natural extension of the said theorems to this sett ing. Ilis results were extended by Hartman and Wintner [3J. Recently a version of Sturm's theorems was discovered by Ahmad and Lazer [lJ for non self-adjoint systems of the ahove type which do not, however, extend the results in the self-adjoint case. The purpose of this note is to present a version of Sturm's theorems which appears to include hoth the self-adjoint and non selfadjoint cases mentioned above. 2. We will assume hereafter that, unless otherwise specified, all matrices P(t), Q(t) are continuous nxn real valued matrix functions whose eigenvalues arc all real functions on I = [a,bJ The points a 7 S in I will be called (mutually) conjugate if there exists a non trivial solution of (1.1) such that y(a)=y(S) = 0 The equation (1.1) will be termed disconjugate on I if I fails to contain any conjugate points, ie. if every non trivial solution of (1.1) vanishes at most once in T. CONJECTURE 1. Let Al (t), A 2 (t), ... , An (t) denote the eigenvalues of QCt). If for each t ( I , maxOl (t), ... , An (t)} ,; 0, 337
(2. 1)
338
ANcrLO H. MINCARELLI
(1.1) is disconjugate on [a,b]. Since we assumed that the eigenvalues of Q(t) are all real, note that
where
Amax{Q(t)}
CONJECTURE Z.
is the largest eigenvalue of Q(t).
Let pet), QCt) be as above. y"
and let Z(t)
~
+
l'(t)y
=
Consider
0,
(Z . Z)
o
(Z . 3)
n be a solution of Z" + QCt)Z
satisfying Z (a)
II
Z (b)
( Z .4)
.
If ,\
max
{P(t)},,'\
-
max
{Q(t)}
( 2 . 5)
for each t ( I , equality not holding everywhere on T, there exists a solution yet) ~ 0 of [2.2) such that yCa) 3.
=
y(c)
=
n
a
(2 .6)
REMARKS 1.
Assume that Q(t) = Q is a constant matrix with real or nonreal entries. Writing (1.1) as a first-order system in 2n-space, a straightforwaru calculation shows that a solution of (1.1) will satisfy yea)
=
yCc) = 0
a
(3.1)
if and only if Q has at least one real positive eigenvalue. Using this it now follows that if
then Q cannot have a real positive eigenvalue hence (1.1) is disconjugate on I. 2.
Let Q(t) be upper-triangular. Writing Q = (qi i (t)), yet) = coI{YI(t), ... , YnCt)}, note that if yet) is a solution of (l.l) which satisfies (3.1) then
339
Sn'R1II 'J'llJ;.OR Y IN N-SV-1Cli
o
(3.
z1
and
Moreover (2.1) wil1 imply that qnn(t)
~
(as qnn(t) is itself an eigenvalue).
The (scalar) Sturm com-
0
each t ( I
parison theorem when applied to (3.2-3) now implies that )'n(t) := n. Now )'n-l satisfies the eqwltion
y"
'n-l
+ lj
n-l,n-l
(t]v
'n-l
Yn-l
+ (j
V
n-l,n'n
(]
o
(c)
Inserting )'n := 0 in the latter shows that )'n- l satisfies an equ3tion similar to (:'i.2). Since (2.1) once more implies qn-l,n-l(t) ~ 0 we agilin find Yn_l(t) - 0 Continuing up the diagonal we eventllally find Y (t) - 0 hence ylt) := 0 l which is impossible. lhus conjecture 1 is also settled in this case. 3.
In the event when pet) := P, Q(t):= Q arc real or complex constant matrices with real/non-reill eigenvalues, conjecture 2 is verified. In this Cilse let ~(.) denote the largest (positive) re31 eigenvalue of a matrix. We can now replace (2.5) by the requirement that ~rp)
>
13.4)
p(Q)
To see this note that we may assume both P, Q are uppertriangular (As there exists a nonsingular T such that P
y
= =
T-IUT where Urt)
is upper-triangular.
The transformation
Tw preserves conjugate points and reduces (2.2) to
1'1" + Uw = O. A simi Iar argument holds for Q). Moreover, we may choose the transforming matrices in such a Wily that ~1(P) (p(Q)) is the first (last) entry of P, Q respectively. Now if 2(t) to satisfies [2.3-4) and 2n(t) to then
2~
+ ~(Q)Zn
= 0
(3. 5)
and
o .
340
ANGELO B. MINGARELU
Set yet) = col{YI (t), 0, ... , O} where YI (t)
t
0 is a solu-
tion of the scalar equation
=
y" + w(P)y
0
(3.6)
We may now apply the comparison theorem to (3.5-6) on account of (3.4). This will show that Yl(a) = yl(c) = 0 for some a
If Zn (t) := 0, then
Thus our Y (t) satisfies (2.6).
Zn-l satisfies the equation
"
0
Zn-l + qn-l,n-lZn-l and Zn_l(a)
O.
Zn-l (b)
If the coefficient is real then
qn-l'n-l ~ w(Q) < W(P)
,
and the argument reduces to the preceding one. On the other if the coefficient is complex then Zn_l(t) := 0, and we.repeat the argument. 4.
Let pet), Q(t) be upper-triangular and assume that Amax{P(t)} = Pll (t),
t
co
[a,b]
In this case conjecture 2 is also verified as an argument similar to the preceding one will show. 4.
APPLICATIONS
In [4J Morse gave the criterion pet) ~ Q(t)
t
co
(4.1)
I
when it is assumed that pet) = P*(t), Q(t) Q*(t) genvalues are real). However (4.1) implies (2.5).
(so that all ei-
Consequently in [3J the symmetry condition on Q(t) was dropped and (4.1) was replaced by pet)
~
QO (t)
t
co
I
(4.2)
where QO(t) = [Q(t) + Q*(t)]/2. They also made use of a result which stated that the disconjugacy of y" + QO(t)y 0 implies the disconjugacy of (1.1), cf.[3J. Now (4.2) implies A {pet)} ~ A {QO(t)} max - max
(4.3)
341
STURM THEOR Y IN N-SPACE
Thus if (1.1) is not disconjugate then y" + QO(t)y = 0 is not disconjugate. Finally (4.3) would now imply that (2.2) is not disconjugate (if the conjecture were valid). Finally in [IJ the following criterion was given: Let p .. (t)
1J
1
<
<
i,
2:
q .. (t)
t
<:
I
(4.4)
2:
0
t
<:
I
(4.5)
1J
n, and as sume tha t q .. (t)
1J
for i ~ j. Suppose further that equality in (4.4) does not hold everywhere in I . Their result is then conjecture 2 with (2.5) replaced by (4.4-5). We will now show that (4.4-5) implies (2.5) under the usual assumptions on P(t), Q(t). To prove this we will make use of a result of Bellman [2, p.294J. By B(t) ~ 0 we will now mean that b .. (t) ~ 0 all i,j. Moreover p(.) will denote the eigenvalue
1J
with largest real part of a matrix. LEMMA. [2, p.294, exercise lJ. a .. > 0 for all i ~ j, then
If B
2:
0 and A is a matrix such that
1J
p(A+B)
~
peA)
(4.6)
.
(Note that the lemma remains valid if we merely assume that a .. ~O foralli~j).
1J
We now set A = Q(t), B = P(t)-Q(t) and note that (4.4) implies B 2 0 while (4.5) implies a .. (t) 2: O. Thus the lemma implies
1J
p(P(t))
-
~
p(Q(t))
tEl
(4.7)
But this is equivalent to (2.5), as we assumed the eigenvalues were real. REFERENCES [lJ
Ahmad, S. and Lazer A.C., An N-dimensional extension of the Sturm separation and comparison theory to a class of nonselfadjoint systems, SIAM J. Math. Anal. 9 (1978) 1137-1150.
[2J
Bellman, R., Introduction to matrix analysis, McGraw-Hill, New York, 1970).
[3J
Hartman, P. and Wintner A., On disconjugate differential systerns, Canad. J. Math 8 (1956) 72-81.
[4J
Morse, M. A generalization of the Sturm separation and comparison theorems in n-space, Math. Ann. 103 (1930) 52-69.
(Second edition,
This page intentionally left blank
This Page Intentionally Left Blank
Spectral Theory of Differential Operators I.W Knowles and R. T. Lewis (eds.) © NorthHolland Publishing Company. 1981
SELFADJOINTNESS OF MATRIX OPERATORS Branko Najman University of Zagreb, Yugoslavia and University of California, Berkeley
Second order differential equations in Hilbert spaces motivate the investigation of matrix operators with unbounded entries. Sufficient conditions are given for the closure of such an operator to be a generator of a C -semigroup or a C -group. The abstract result is a~plied to the KleiR-Gordon equation MOTIVATION Consider the differential equation 2 d u(t) _ qdatt) + Hu(t) = 0 dt 2 in a Hilbert space Cl • Setting
vet)
B
=IU(t)] ~att)
bH
:J
-6
-
~(O) = u 1
u(o)
(0.1)
u( t ) -
wet) = datt) - qu(t)
j
, A
we have formally
datt) = Av(t) datt) = Bw(t) (0.2) with appropriate initial conditions. If -H is an elliptic differentitia1 operator, q a function, (0.1) is a hyperbolic equation. We assume ~na~ H is bounded from below, Re q bounded from above. We shall find sufficient conditions on q, H in order that the closures of A, B generate C -semigroups in natural Hilbert spaces associated with (0.2). 0 Obviously (0.1) includes many physical systems with friction ([~). Another example is ~he K~ein-Gordon equation; in this case q is skew-symmetric : q = iq, q is a real-valued function, H a Schrodinger operator. Under our conditions the closures of A, B generate Co-groups. If H is positive definite, this reduces to the familiar conclusion : iA , iB are essentially selfadjoint. 1. General results Let ~ be a Hilbert spac~. H a symmetric operator in ~, bounded from below by a. Let H be the Friedrichs extension of H , T = H + 1 - a • Denote by ~1 the domain of the positive square /2. root T1of T, equl.pped with the norm u ~ I T1/2 uII ; by ~-1 the completion of ~ wi th respect to the norm u ~ I T -1/2ull , be the extension of 343
H
to
I"('l -J"I 'I.) '3-1 •
.Iv
344
BRANKO NAJMAN
For a densely defined operator q in ~ we define operator A in '1t-1 • B in Xe : ~(A) .2) (H) @(.1)(q)n~1) AU!9v=v!9 (-Hu+ qv) tl (B) (.tl (q) n ~1) !9lfaBu !9 v = (qu+v) !9 -H' u Define Q T- 1/ 2 qT- 1/ 2 . The following three assumptions will be of interest: (Q) (QB)
Q is a closable densely defined operator Q is a bounded densely defined operator qT- 1/ 2 is a bounded densely defined operator
(aN)
Obviously (QN) ~ (QB) ~ (Q) • (QN) was assumed in [5] • (QB) in [3]. A sufficient condition for (Q) is that q is symmetric or skew-symmetric, ~(q)n tJ (iil is a core of H • A sufficient condition for (QB) is that there is a core D of the form h of H and ~ > 0 , a real such that h (u) - ~ I (qu Iu) I .:: a I u II 2 uED ( 1 • 1 ). If h(u)
wi th
D, a •
~
~lIqull
-
as in
2
.:: allull
2
(1.1), then
q
uED has an extension
(1.2).
qE,U~~I~) Theorem 1. a) Assume (Q). Then A and B are densely defined closable operators. If Q is closed then B is closed. If H is selfadjoint, Q closed, then A is closed. b) Assume (QB). Then the resolvent set (B) of the closure of B is nonempty. Moreover, if D1 c lJ (q)r1 is dense in ~1 , D2 dense in ~ ,then D1 !9 D2 is a core of ~. If H is essentially selfadjoint then p(I) is nonempty. c) Assume (QB) and there are real numbers B such that
1-1
°.
I
(Hu u)
>
-
0 II ull 2 :2
U E
r;n (H)
Re (qu \ u) ~ Sll u \I UE JJ (q) Denote by Ao the larger root of the equation
(1.3)
(1.4).
A2 - AS - 0 = 0 (1.5) ( 0_ = max {O, - o}). Then n generate~ a C -semigroup of type not larger than Ao • The same holds for A if 0 H is essentially selfadjoint. In particular if H is a positive definite essentially selfadjoint operator, q Skew-symmetric, then A and B are essentially skew-selfadjoint. Theorem 2. Assume (aN). Then a) .l)(I) = J)(lJ) !9 ~1 IU!9 v = v !9 -Ru + gv) = ~1 (!) ~ ~ !9 v = (gu + v) !9 - H'u b) Assume (aN), (1.4) and (1.5) hold, Ao is the larger root of (1.5). Denote CO ={A~ Re A > Ao }' The following statements are
.pon
345
SE LFADJOINTNLSS OF MATRIX OPliRATORS
equivalent : (i) !t(A- A) is dense for some A E C 0 (ii) ~(A- A) is dense for all A E C 0 (iii) 9.-(H - \q + A2) is dense for some \ E \ E (iv) ~(H - Aq + A2) is dense for all (v) H is essentially selfadjoint.
C C
0 0
We sketch the proof of Theorem 1b) and Theorem 2b) (the rest is rather standard). Proof of Theorem 1b} • It is easy to see that it is sufficient to consider positive definite H and set T = H. We prove only the statement concerning A; the other part is easier. By (QB) q can be extended to q E :£ ( ~11 ~ -1) • Define _ _ ":;D(A) ={U Ell VE;!t1 :vE~1 ' UEH,-l~v+JJ(H) Au Ell v_~lv Ell (-H'u+ qv)}. It is easy to see that A is injective, A E Je (';1(,1 ) • We have to prove that if H is essentially selfadjoint then A = A. _ Let UEil v E X) (A). Then v E~1 , U = U + E,-lqv where U Eel> (H). Choose vn E.p(q)n ifa,1 ' un E~(H) such that vn -----+ v in ~1 (H+1) un -..;. (H+l)u , in ~(we used (QB». Since ~(H) is dense in ~ we can find un E J:J (H) such that II-Ho. + qv I < _- 1 , --1 n n Set wn = H (-HUn +qv n ) -un + H qv n · Then wn ->- 0 , Hw - - + 0 in ,/, • Let un U + U • Obviously u E J) (H) and n a n n n
*.
A
1 im
n->-oo
U
n
in ~ hence find
=
1 im
n+oo
U
n
+ l'
A
n!: un - -1,
=
(note that H' q E __ -HUn = qV n --+ -Hu Hl / 2 + H,-lqv)
(_u n
1.
(
U - n~~
- -1
wn - H qv n )
=
-
- , -1,
U + H
qv
=
U
t ( "'1) 'J
) • Further HUn Hu - HW n + qVn' - 1/2 n in ~. From H w --+ 0 in ~ we = El/2~ + El/ 2 (H,-lq?v - H,-l CfI)-----+ 0 n
n
in ~ • Since un converges to U in ~1 we conclude that un converges to u + H,-lqv = u in ~1 • Thus we have found zn = un Ell vn E JJ (A) such that zn --+ u Ell v AZn ---+ v Ell - Hu = AZ in 'Jt 1 (i) , (ii) and (v) are equivalent by an eorem 2a) We shall prove that if A E Co then the range of A - \ is dense iff the range of H( A ) = H - A q + A2 is dense. This obviously implies the equivalence of (i) - (v) • Assume Z = x Ell Y 1.E.'9.. (A-A). By Theorem 1c) Co C p(1\.) (A as in the proof of Theorem 1b) ), therefore Z It(A- A). Let U E;D(H) = JJ (H(A)) then U Ell AU E!l(I) by Theorem 2a) and 0 = (zl(A-A)u Ell AU) = -(y/H(A)U), hence y is orthogonal to 5t(H(A)) . • Also 0 Ell v EJ)(I) for every VEJ) (E) and (zi(A-\)OEllv) =0 = (T 1 / 2 xI T1/2v) + (y(q-A)V) = (x + [(q_\)T-lj\\TV). We conclude
346
lJRANKO NAJMAN
YE ,1.(H(!..)) .1
-[(q- A)T -1 ]
X =
Conversely, if x, y satisfy (1.6), then orthogonal to ~ (A- A).
z
*y
=
x
( 1 •6)• 6)
Y E J(,1
is
As regards Theorem 1c) • note that if q is skew-selfadjoint, then -A, -B are similar to A' , B' ; A' • B' are A, B with q replaced by -q • Thus A, B generate Co-groups. 2. Application We apply the preceding results to the case ~= L2(Rm) • , H is the Schrodinger operator, q a multipli~ation operator. 2
m
Set
I
(-io.-a.) j=l J J
L=
, where
+V
m a j E Lioc(R ) , div a th~
negative part V_ of V is 6
less than
1 :
lim sup
~-+o YERm
(see
L2loc (Rm) , VEL2loc (Rm) (2.1 ), -form bounded with relative bound E
J V (x-y) x 2-m dx = 0
Ix I < a
( 2.2)
-
A sufficient condition for (2.2) is that V is a sum of Vi • Vi E LPi(Rm) , Pi > ~ • Let 7J (H) = Co'" (Rm) • Hu = Lu for u (H). Then H is essentially selfadjoint (see [41 for V = 0 , the general case follows easily from this). _ As in Section 1 , we choose ex ~ 0 such that T = H + a is posi ti ve definite and define the spaces ~ 1 • ~ -1 • ';}t1 , ';}to usi ng T. Note that ~-1 and -ae,o are distribution spaces, H': ~1 -+ ~-1 is a differential operator (this follows from the fact that C "'(Rm) is dense in ~1 • i. e. it is a form core of H - see [7]) •• 0 Now assume that q is the operator of multiplication by the function iq , {J:) (q) = C; (Rm) • q is a real valued locally square integrable function such that q = q1 + q2 • Iql (x) I ~ C V+(x) (2.3) for some C ~ 0 • V+ = V + V_ , [6]
).
{ I q2 (x-y) II x 12 - m yERm Ix <1 sup
or
sup yERm
J
I x 1<1
Iq2 (x-y)
Define A in 'Je 1 , B Au 6) v = v 6) (-Lu+qv) • ,{D (B) = (qu+v) Ell -Lu •
dx
( 2.4)
<
00
2 121 x 1 - m dx
<
'"
( 2.5).
m
Co'" (R )
00
6)
m
Co (R ),
,Bue v =
Proposition ~. Assume (2.1), (2.2) • (2.3) and (2.4) hold. Then (QB)olds. A and B are closable densely defined operators. The closures A • ~ are distribution operators; they generate C groups. If H is positive definite and T = H • then A. B a~e skew-selfadjoint.
347
SEL!',lDjOINTNESS Of' ;\lATRIX OPERATORS
This follows directly from Theorem 1, since (2.3) and (2.4) (QB) (by [6], (1.1) balds) • If (2.5) holds, then (1.2) is satisfied, thus (QN) holds, so more precise statements can be made using Theorem 2 • We shall do it in the more special case of Klein-Gordon operator. In this case Vex) = m2 - q(x)2 • To satisfy (2.2) - (2.5) we have to assume that q satis£ies (2.5) • i.e. q1 = 0 • Indeed, since V+ is bounded, (2.3) implies (2.4) and (2.5). There£ore we can set q1 = 0 • q2 = q • Since V must satisfy (2.2) we conclude that
imply
q
must satisfy lim sup
8~o yERm
Ix
lij(x-y) 12 1x1 2 - m
{
dx
=
O.
(2.6) •
<6
In other words, the assumption (2.2), necessary to ensure the essential selfadjointness of H automatically implies that (QN) is satisfied. Of course, Theorems 1 and 2 apply even if (2.6) is not satisfied. It can happen however, that 4l COO is not a core 0 f o a generator of Co-group if (2.6) does not hold.
3. Extensions The results of Section 2 are obtained from a general theory. We can expect stronger results in the s?ecial case from a direct treatment. This is indeed the case; (2.4) can be dropped in Proposition 3. Propos! tion 4. Assume (2.1), (2.2) and q L~oc (Rm) • Then A, ~
are distribution operators generating Co-groups. If H is pOSi ti ve definite and T = H , then A, ~ are skew-selfadjoin t • The proof is similar to the proofs in Section 1, using the results on Schrodinger operators with complex potentials (contained in [1) [2] i f a = 0 ) • j Even (2.1) can be relaxed - it is sufficient that I
a. E I.?
(Rm)
VEL1
( Rm)
(2.1). ' loc m Namely i f (2.1). (2.2) hold together with q Soc{R ) ,then A , defined on J) (Al = (j) Oil 4l (Rm) by Au 4l v = v Qj (-flu + qv) is seen to be closable, the closure being a generator of a Co-group. 2 (Rm) We can go even further - by relaxing q. E Lloe to q+ or q_ is t. -form bounded J
-.Loc
c:
I
or
q E i{oc(lfl) , p = ~ By defining the maximal operators ~ax case of Schrodinger operators, we can show that generators of Co-groups.
Bmax as in the Amax • Bmax ar e
348
~J
[2J [3J [4J
[51 [6J [7J
BRANKO NAJMAN
REFERENCES Brezis,H.,Kato,T.,Remarks on the Schrodinger operators with singular complex potentials,J.Math.Pures Appl.58(1979)137-151. Kato,T.,On some Schradinger operators with a singular complex potential,Ann.Sc.Norm.Sup.Pisa,Ser.IV,5(1978)105-114. Krein,M.G.,Langer,H.,On some mathematical principles in the linear theory of damped oscilations of continua, Integral Eq.and Op.Th.l(1978). Leinfelder,H. ,Simader,C.G. ,Schrodinger operators with singular magnetic vector potentials,preprint,Bayreuth 1980. Najman,B.,Solution of a differential equation in a scale of spaces,Glasnik Mat.,14(34)(1979)119-127. Schechter,M.,Spectra of partial differential operators,North Holland 1971. Simon,B.,Maximal and minimal Schrodinger forms,J.Op.Th.1(1979) 37-47.
Spectral Theory of Differential Operators I.W. Knowles and R. T. Lewis (eds.) © North·Holland Publishing Company, 1981
SPECTRAL PROPERTIES OF SOME NONSELFADJOINT OPERATORS AND SOME APPLICATIONS* A. G. Ramm
Mathematics Department University of Michigan Ann Arbor, Michigan
Let A be a compact linear operator on a Hilbert space H, s (A) n
=
{A (A*A)}1/2, Q be a compact linear operator, 1+ Q ~
n
invertible, B
A(I+Q). We prove that sn(B)sn (A) + 1 as n r a 1 a If IQfl i cIAfl lfI - , a > 0, c > 0, feH and sn(A) = c n1 q Y {1+0(n- )} r,q > 0, then s (B) = s (A) {1+0(n- )j, where =
-1
n
+ ~.
n
Y = min {q, ra(l+ra) }. This estimate is close to sharp. We also give conditions sufficient for the root system of B to form a Riesz basis with brackets of H. Applications to elliptic boundary value problems and to some scattering problems are given. NOTATIONS, DEFINITIONS Let H be a separable Hilbert space, A and Q be compact linear operators on H,B
=
=
{A (A*A)}1/2 n be the s-values of A (singular values of A), c be various positive constants, A(I+Q), AnCA) be the eigenvalues of A, snCA)
An{(A*A)1/2}
Rd be the Euclidean d-dimensional space, D C Rd be a bounded domain with a smooth 2 boundary, L be a positive definite in L (D) elliptic operator of order 2 and M be a nonselfadjoint differential operator of order m < 2. We define s (L) = l {s (L- )}-l. Let A~ = A~, ~ I O. With the pair (A,~) one associat:s the Jordan n
chain defined as follows: consider (*) A~(l) - A~(l) =~. If this equation is not solvable then one says that there are no root vectors associated with the
(A,~). If C*) is solvable then consider the equations (**) A~(j) - A~(j) = ~(j-1), j = 1,2, ••. ,~(0) =~. It is known [1], that if A is compact then there
pair
exists an integer N such that (**) will not be solvable for j
> N.
In this case
~(l) , ••• ;~(N) are called the root vectors associated with the pair (A,~), (~,~(1) , ••. ,~(N» is called the Jordan chain associated with the pair (A,~). Con-
vectors
sider the eigenvectors
~l""'~q
corresponding to the eigenvalue A and all the
root vectors associated with the pairs
(A,~
p
), p
= 1, ••• ,q.
The linear span of
the eigen and root vectors corresponding to A is called the root space corresponding to A. The collection of all eigen and root vectors of A is called its root system. Let us define Riesz's basis of H with brackets. Let {f.} be a linearly J
independent system of elements of H, {h.} be an orthonormal basis of H, and J
m < m <•.• mj + ~ be a sequence of integers. Let Hj(F ) be the linear span of l j 2 vectors h .h +l, ••• ,h -1' (f ••••• fm -1)' T be a linear bounded invertible m m m m j j j+1 j j+1 operator from H onto H, TF. = H .• j 1.2, •••• Then the system {f.} is called a J
Riesz basis of H with brackets.
J
If m
=
*Supported by AFOSR 800204 AMS Classification 47A55, 47AlO, 35 P 20 349
j
then
{~}
J
is called a Riesz basis of
350
A.J. RAMM
II. If a root system of A forms a Riesz basis of H wi th brackets then we write AERb (I!). If i t forms a Riesz basis then we write AE:R(Il). The range of A is denoted by RCA), lim means lim as n + =, N(A) = Ker A the set consisting of the zero element of H.
{~:
=
A~ =
OJ,
to}
denotes
INTRODUCTION Two questions will be discussed:
1) When is Sn(B) - sn(A) and what is the ordec
2) When does BE:~ (II)?
of the remainder?
There are few known results connected
with question 1). The results are due to H. \oIeyl, Ky Fan and M. G. Krein (see [2]), and the author [3]. It seems that there were no abstract results on the perturbations preserving asymptotics of spectrum with estimates of the remainder. In Theorem 1 (Section 3 below) such a result is given. In [2] there are some results about completeness of the root systems of certain operators. In Theorem 2 an abstract result which gives an answer to question 2) is given. In Theoreln 3 some spectral properties of nonselfadjoint elliptic operators are presented. F. Browder [1, Ch. 14, Theorem 28] proved completeness of the root system of L + H 2 in H = L (D). He prove that L + HER (l!) by applying Theorem 2. In order to do b 1 this note that (L+H)-l = A(I+Q), where A = L- , Q = _(I+HL- 1 )-lML- 1 . During the last decade there was a great interest among physicists and engineers in question 2) and some results due to t1arkus, Kacnelson, Agranovich and others were used [4] (see also Appendix 10 in [3] and [5]). RESULTS He will not repeat in this section the notations and assumptions of Section 1 but they are assumed to be valid. THEOREt! 1. IQfl sn (8)
< clAfl
-
If N( 1+Q) = {O}, dim R(A) = "', then lim s (B) s -1 CA) = 1.
a
If I
I-a
sn (A) {l+O( n
=
, a > 0, for all fEH and s (A) _y n
)}, where y
=
n
n
If
_
> 0,
cn- r {l+O(n q)}, r,q
=
-1
min {q, raO +ra)
then
}.
REl1ARK 1.
The estimate of the remainder is close to sharp: for the elliptic op2 erators in L (D) the remainder is of order given in Theorem 1. r THEOREH 2. I f A> 0, \(A) - cn- as n + "', r > 0, IQfl i clAafl, 0 < a,
N(I+Q)
=
{O}, and ra
TI!EOREH 3.
If t -
to}, then sn(L+H) REHARK 2. If d then L+MER (H). b
ffi
~
1, then BERb(H).
~
d then L + M£Rb(H) , H
sn(L) {l+O(n
1 then m
-y
)}, where y
implies
~
- m
2 L (D).
=
min {d
=
> 1.
Furthermore if N(L+H) -1
, (t-m)(t-m+d)
Therefore if d
=
-1
}.
1 and m
<~
THEOREH 4. If the assumptions of Theorem 2 hold and ra > 2 then the equiconvergence of the eigenvector expansion for the operator A and the ~oot vector expansion with brackets for the operator B holds. REHARK 3. The meaning of the equiconvergence is as follows. Let g be an arbitrary element of H, {h.}, j = l, 2, ..• , be the system of eignvec:tors of A which ]
forms an orthonormal basis of H, {h.} be the root system of B. J
a sequence of integers m].,
ffi].
+
00
such that I I
Then there exists
n
L (P .-p~)gl I
j=1
]
]
+
0 as n
+
00.
Here
speCTRAL PROPER'J'JES OJ' S(),\lF NONSFLFADjOINT OPliR.'lTORS
P.(P~) J
J
is the projection onto the subspace F,(H.) defined in §l. J
J
351
That is equi-
convergence means that the eigenvector expansions and the root vector expansions with brackets converge or diverge simultaneously. For the first time the equiconvergence theorem for the Fourier series and for the eigenfunction expansions for a regular selfadjoint Sturm-Liouville operator was proved by A. Haar (1910) and M. Stone (1928). Since then there were many results in this field but they were obtained for selfadjoint differential operators and in most cases are based on some study of the asymptotics of spectral functions of these operators [6]. The above result is of abstract nature and deals with nonselfadjoint operators. APPLICATIONS Consider the following scattering problem: 2 +k ) u = 0 in [I, u = 0 on S, u = u + v,
('i
(1)
k > 0,
o
DC R3 is a compact domain with a smooth boundary S,
where
v satisfies the radiation condition,
[I
U
3 = R 'D.
exp {ik(n,x)},
o
If one looks for the solution of (l) of the form v = Jsg(x,t)f(t)dt, -1
g = exp(iklx-tl)(4TIlx-tl) , then Af = -u ' where Af = Jsg(s,t)fdt, SES. The O 2 operator A on H = L (S) is nonselfadjoint. Its spectral properties are of interest [3]. One can use theorems 1, 2 for studying these properties. Consider the problem 2
2
3
[V+k-q(x)]u=OinR, with the same u
o
u=uO+v,
k>O,
(2)
as above and with v satisfying the radiation condition.
that q(x) is compactly supported, q(x) = 0 if Ixl > R, qEC~. tion for u is u = Uo - Jg(x,y)q(y)u(y)dy
= Uo
J=
- Tu, 2
The integral equa-
JIYI~R'
Here T is a compact operator on II = L (DR)' DR per ties are of interest.
Assume
(3 ) {x:
Ixl
< R}.
-
Its spectral pro-
2
Namely i t is of interest of know i f TERb(H), H = L (DR)'
PROBLEMS 1)
1
2
2
Let Bf = Ll exp {i(x-y) }Edy be an operator on II = L ([-1,1).
known i f
BE~(H).
It is not
2) If d > 1 i t seems to be an open problem i f L + MER(H) under
the assumption of Theorem 2. Is the bracketing necessary? Some other problems can be found in [3) and [5], where some questions of interests in applications are also discussed. SKETCHES OF SOME PROOFS 1)
Theorem 1.
Let U = A*A, V = B*B = (I+Q*)A*A(I+Q), Ln be the linear span of
(I+Q)f and Mn = (I+Q*)L • Then the condition glL n n is equivalent to f 111n' where 1 means the orthogonality in H. From the minimax
n first eigenvectors of U, g
princple it follows that
352
A.j. RAMM
< sup -
(Vf,f)
If1 2
flL
(Ug,g) < sup cg;g)
-
f1M
fllr n
.;~ m
(g,g) I;j2 -
n
2 (A) {I + sup sn+1 flL m (4 )
(I+Q)-1, S
Taking into account that U ; (I+S*)V(I+S), where 1+ S we conclude that 2 2 a sn+1+2m(A) i sn+1+m(B) {I + O(sm(B»}. From (4) and (5) it follows that s (B)s-l(A) n
2 sn+1+2m(A)
s
S~+l+m(A)
m
(A)
n+m Sn(A)
n+m
provided that mn s
(B)
n+1+m sn+l+m(A) = 1
=
-1
+
O.
+
00, and
<
(6 )
sn+l+m(A)
r
q
{I + O(n- )} implies that
{I + O(n- q )}
= (__n__ )r
as n
+ 1
(5)
(B)
2
-
The assumption sn(A) ; cns
2
{I + O(sa(A»} < n+l+m
n
_(I+Q)-lQ,
=
Let mn
{1 + O(~) + O(n- q )} n+m
-l-x n
x
> O.
(7)
Then (6) and (7) imply that
1 + 0 (n-(l-x)ra) + O(n- q ) + O(n-x)
+ O(n- Y),
(8)
where
Y = min {q,(l-x)ra,x}
min {q,ra(l+ra)
-1
}.
(9 )
Theorem 1 is proved. It is known [7], that -1
-1
d2 2 N(A) = cA (l + O( A)} (10) where N(A) is the number of the eigenvalues An of an elliptic selfadjoint opera2 d tor L on II = L (D), D C R , ord L = 2. Thus td- l _d- 1 An ; c n {1 + O(n )}, (11) 1
because An is the inverse function with respect to N(A). remainder in (10), and therefore in (11), is sharp. (L+M)
-1
=
A(I+Q) , Q = -(I+T)
sn(A) {I + O(n a
= (2-m)2-
and 2 - m O(n28].
d
-1
1
-Y
-1
)}, Y = min {d
T, T ; ML -1
2-m '2-m+d}.
, and we used formula (9).
> d(d-1)-1
-1
,A
=
-1
L
The estimate of the
For the operator B; Theorem 1 says:
In this case
Therefore Y
=
I' ;
2d
-1
2-m 2-m+d if d
sn(B)
,q =
1.
If d
>1
then the estimate of the remainder given in Theorem 1 is
-
) and it is sharp.
The first statement of Theorem 1 was proved in [3, p.
353
SPECTRAL PROPERTIES OF SOME NONSELFADjOIN'j' OPliRA'fORS
2) Theorems 2, 4. In [3], Appendix 11 the following proposition was proved: assume that L > 0 is an operator on a Hilbert space H with a discrete spectrum r
l
), r < r, r > 0, and ITfl ~ clLafl, a < 1, where T is a linear l (nonselfadjoint) operator; i f r(l-a) ~ 1 then L + TS~(H), i f r(l-a) ~ 2, then Aj
=
cjr + O(j
the eigenvector expansion for the operator L and the root vector expansIon for the operator L + Tare equiconvergent. then BSRb(H).
Let A-I = L, B-
1
= L + T.
He have B = (L+T)-l = A(I+TL-l)-l = A(I+Q) , Q
The operator (1+TL
-1 -1
)
is bounded.
I f B-ISRb(H)
l _(I+TL- )-l TL -l. a
Therefore the inequality IQf I ~ c I A f I im-
plies that ITL-il < cIL-afl, or ITfl < CILl-afl. Thus a = I - a and the conditions r(l-a) ~ 1, r(l-a) ~ 2 are equivalent to ra ~ 1, ra ~ 2 respectively. 3) Theorem 3. The argument given after formula (10) proves the second statement of Theorem 3. The first statement of this theorem follows from Theorem 2. In1
deed r = £d- , a = (£_m)£-l and the condition 1 < ra can be written as £ - m> d. This condition implies that L + NSRb(I{). 4) Applications to scattering theory. The operator A defined in n.4, can be written as A = Al + iA2 = ReA + iImA, where Al (A+A*)/2, A2 = (A-A*)/(2i). The kernel of Al is cos {kl s-t I} (411 I s-t I) -1, k > 0, while the kernel of A2 is sin (k I s-ti ) (4111 s-t I) -1.
The operator Al is an elliptic pseudo-differential op-
erator of order -1, while A2 has the order -00: -1
ator.
-1
Suppose that Al
IQfl ~ clA~fl with a mate).
and A
<1
exist.
it is infinitely smoothing oper-
Then A = Al (I+Q), Q
(actually a can be any number
-00
-1
iA l
A2 and in this esti-
Also Al is not necessarily positive it can have only a finite number of
negative eigenvalues. 2
Ii = L Cf).
Therefore Theorem 2 is applicable and says that ASRbCH),
Let A be invertible.
If Al is not invertible then it has a finite00
dimensional null space and if SeC then the elements of the null space are in C . Therefore one can add to Al a finite dimensional operator of order -00 and get an invertible operator, and subtract this finite dimensional operator from iA2 without changing its order -00. After this operation the above argument shows that AeRb(H). If A is not invertible but H can be decomposed into a direct sum lIO + III where liO is the finite dimensional root space of A corresponding to the eigenvalue 0, and
~
is invariant space for A in which the restriction A(1) of (1)
A on III is invertible, then A
SRbCH ) and ASRbCH). Spectral properties of some l operators arising in diffraction theory were studied in [4]. Completeness of the root system of the operator A defined in n.4 was first proved in [8]. REFERENCES [1]
Dunford, N., Schwartz, J., Linear operators, Vol. 2, (Interscience, New York, 1963) .
[2]
Gohberg, I. C" Krein, M. G., Introduction to the theory of linear nonselfadjoint operators, (AMS, Providence, 1969).
354
/II
R/lMM
[3)
Ramm, A. G., Theory and applications of some new classes of integral equations, (Springer Verlag, New York, 1980).
[4)
Voi tovich, V" Kacenelenbaum, B., Si vov, A., Generalized method of eigennoscillations in diffraction theory, (Nauka, Moscow, 1977) (Russian).
[5)
Ramm, A. G., Mathematical foundations of the singularity and eigenmode expansion methods (SEM and EEM), (to appear).
[6)
Levitan, B. M., Sargsjan, I. S., Introduction to the spectral theory: adjoint ordinary differential operators, (AMS, Providence, 1975).
[7)
H~rmander, L., The spectral function of an elliptic operator, Act a math., 121, (1968), 193-218.
[8J
Ramm, A. G., Eigenfunction expansion corresponding to the discrete spectrum, Rad. eng. elect. phys., 18, (1973), 364-369; M. Rev. 50 #1641.
self-
Spectral Theory of Differential Operators I. Knowles and R. T. Lewis (eds.) © North-Holland Publishing Company, 1981
w:
DIRICHLET SOLUTIONS OF FOURTH ORDER DIFFERENTIAL EQUATIONS Thomas T. Read Western Washington University Bellingham, Washington 98225
°
The equation y(4) - (PlY')' + poy = has exactly two linearly independent solutions on [0,=) with finite Dirichlet integral.
Some applications to the determi-
nation of the domains of self-adjoint operators associated with the differential expression and to the minimization of a quadratic functional are discussed. Let L be the fourth order differential expression defined by L[y] on [0,=).
=
y(4) - (PlY')' + poy
For simplicity we assume that PO is continuous and that
PI is continuously differentiable. lution of L[f]
=
°
We shall call f a Dirichlet so-
if
f~(lf"12 +
P l lf'12
+ p o lfl 2 )
<
=.
°
THEOREM 1. Let PI ~ 0, Po ~ c > on [0,=). Then there are exactly two linearly independent Dirichlet solutions of L[y] = 0. The corresponding deficiency index problem--that of determining the number of solutions of L[y]
=
° which
are square integrable on [0,=)
when L has nonnegative coefficients--is still open, although it is known that the number of square integrable solutions is two under various additional conditions on the coefficients. instance,
[2]-[10],
[12],
[15].)
(See, for
We shall see below, however, that
determining the number of Dirichlet solutions can suffice for many purposes. For higher order expressions, an example has been given by Kauffman [12] of a sixth order equation with positive polynomial coefficients which has four square integrable solutions.
The question of whether
the number of Dirichlet solutions is always half the order of the expression remains open.
Kauffman [13] showed that the answer is
affirmative provided that the coefficients are reasonably regular, 355
356
THOMAS 1'. RHAD
and his results have been improved somewhat by Robinette [14].
How-
ever Theorem 1 is the only result to require no restriction on the coefficients beyond positivity. A general plan of the proof runs as follows.
The general theory of
the Dirichlet index (Kauffman [13J, see also Bradley, Hinton, and Kauffman [l])implies that the number of Dirichlet solutions is at least two.
If it exceeds two, then there is a real-valued function
f such that L[f] = 0, f(O) = f' (0) = 0, and
f~ ( If" 12
+ PI I f' 12 + PO I f 12)
=
1.
Then for each positive x,
o
f~ L[f]f= _(f[3]f+ f"f')(t)lo+f~(lf"12 + Pllf' 12 + PolfI2),
=
where f[3] denotes the third quasi-derivative of f.
Thus D(x)
= f[3] (x)f(x) + f"(x)f' (x) increases monotonically to one.
The
second term in D(x) is in Ll(O,oo) so that the first, DI(x), must actually be close to one most of the time.
A careful analysis. of
the behavior of f and its derivatives reveals that f, and hence also Dl , is oscillatory (this was first established by Hinton [10]), and that Dl is bounded above by one. Thus the graph of Dl consists of nearly horizontal segments separated by very short intervals on which Dl decreases to zero.
This situation can be exploited to pro-
duce a contradiction. Theorem 1 can be extended to cover many expressions with a nonconstant leading coefficient P2.
In particular, the inequality
P2' :: - KP 2 is sufficient.
(1)
However it is not known whether the result holds for
an arbitrary leading coefficient.
The argument sketched above does
not seem to extend readily to this situation except under some hypothesis such as (1). Theorem 1 leads to a simple explicit characterization of the domains of certain self-adjoint operators associated with L. f(O)
We shall write
(f(O), f'(O), f"(O), f"'(O»and, for functions with finite 2 Dirichlet integral, (f,f)n = f (lf"1 + Pllf' 12 + p o lfI 2 ). Let S be =
o
any two dimensional subspace of ~4 with the property that if f and g are in CO[O,m), the class of restrictions to [0,00) of·~O(R) functions, with f(O)ES, g(O)ES,
DIRICHLET SOLUrIONS
(Jj-
357
/;OURHI ORDJ:'R DIFFERl'NTHL FQUATIONS
then
Note that S is simply any two dimensional subspace which defines symmetric boundary conditions at O.
Thus the operator TS defined by
TSf = L[f] on domain TS = {fEC~[O,=) is symmetric.
: f(O)ES)
TS can be seen to be bounded below, so that it can be
made larger than a positive multiple of the identity by adding a suitable positive constant to PO' THEOREM 2.
Let TS be ~ above and suppose TS ~ cI, c > O.
Then the
Friedrichs extension" HS ' of TS satisfies domain HS = {fEdomain ~ax
(f,f)D <
domain H§/2 = {f : f'EAC loc '
(f,f)D < =
00,
r(O)ES), (f(D),f' (D»ETT 2 S).
Here Lmax is the maximal operator associated with Land TTZS is the projection of S on its first two components. This result should be compared with the following theorem of Kauffman [13] for Zn-th order expressions L[y] = Z~_O(-l)j(P.y(j»(j). We extend the notation used above by writingJf(O) = n I f (j)1 2 . ( f , f )D -- fO= Lj=OPj THEOREM 3.
(Kauffman).
Pj ~ 0 and Po ~ (a)
E >
(f~O),
... ,f(Zn-l) (0»
Let L[y] = Tj"=D(-l)j (Pjy(j»
0, Pn ~ ~ > D.
and
(j) with each
If
S is an n dimensional subspace of C2n such that
f, gEC~[O,oo), f(O)E S, g(O)E S ~
f~ L[f]~ = f~ ~~=OPjf(j)~(j), and (b)
(2 )
there are exactly n linearly independent Dirichlet so-
lutions of L[y] = 0, then the Friedrichs extension HS of the positive symmetric. operator TS with domain {fECoo[D,oo) ---c
: f(O) ES} satisfies
domain HS
{f
domain Hl/2 S
{f:f(n-l)E AC loc ' (f,f)D < 00, TInf(O)E TInS}.
E
domain Lmax:
(f, f) D
< 00,
f (0) E S},
This result has been extended to a wider class of expressions in a weighted Hilbert space by Bradley, Hinton, and Kauffman [1].
The
characterization of domain H~/2 was obtained earlier by Hinton [11] under the assumption that L have exactly n square integrable solutions.
It is also shown in [1] and [13] that if condition (b)
358
THOMAS T. READ
fails, then the domains of HS and
H~/2
are proper subsets of the
indicated sets. Condition (a) of Theorem 3 is a restriction to a certain class of symmetric boundary conditions.
It can be seen to be unnecessary by
showing that for any n dimensional symmetric boundary space S the difference between the two sides of (2) can be written as a quadratic form in
TI
n
f(O) and
TI
n
g(O) and so can be estimated in terms of
arbitrarily small multiples of (f,f)D and (g,g)D and some mUltiples of the L2 norms of f and g. Thus the results of (1) and (13) actually hold for arbitrary symmetric boundary conditions. Condition (b) in Theorem 3 is certainly satisfied in the context of Theorem 2, for it is precisely the assertion of Theorem 1.
Thus
Theorem 2 can be proved in the same way as Theorem 3, with the only alterations necessary being those required to accommodate arbitrary symmetric boundary conditions.
H~/2
One application of the characterization of the domain of
in
Theorem 2, discussed at length in (1), is to the minimization of the quadratic functional Q(f) over the set
=
1;(lf"1
2
+
P l lf'I
2
+
POlf12)
~ of all functions f with L2 norm one for which the
Dirichlet integral is defined and finite, and the boundary values (f(O), f' (0)) lie in some specified subspace So of 2,
~
H~/2
~2.
By Theorem
is precisely the set of elements of unit norm in the domain of where S is any two dimensional symmetric subspace of
C4 such
TI S = SO. Thus if S satisfies condition (a) of Theorem 3, then 2 the infimum of Q(f) for f E~ is the least point of the spectrum of
that HS·
It follows that the infimum of Q(f) may be determined from the elements of C~[O,oo) satisfying f(O)E S, that is, from the domain of the symmetric operator TS of Theorem 2. boundary condition is f(O)
In particular, when the
= f' (0) = 0, then the infimum of Q(f)
over ~ is equal to ·the infimum of Q(f) over the C'Q[O,oo) functions with unit norm.
It should be emphasized that this equivalence
depends on the result of Theorem 1, for if the equation L[y) = more than two Dirichlet solutions, then
~
°
had
would include functions
not in the domain of H~/2, and the infimum of Q(f) over ~ might be less than the infimum of Q(f) over the elements of domain TS of unit norm.
DlRTCHLET SOLUTI()NS OF [(){ 'RTII ORiJliR DIFFERENTI:lL EQ(,':lTIONS
359
REFERENCES: [1)
Bradley, J.S., Hinton, D.B., and Kauffman, R.M., On the minimization of singular quadratic functionals, Proc. Royal Soc. Edinburgh, to appear.
[2)
Devinatz, A., On limit-2 fourth order differential operators. J. London Math. Soc. (2) 7 (1973) 135-146. Devinatz, A., Positive definite fourth order differential operators, J. London Math. Soc. (2) 6 (1973) 412-416.
[3) [4) [5) [6)
[7) [8]
[9] [10)
[11)
Eastham, M.S.P., On the L2 classification of fourth-order differential equations, ~r. London Math. Soc. (2) 3 (1971) 297-300. Eastham, M.S.P., The limit-2 case of fourth-order differential equations, Quart. J. Math. 22 (1971) 131-134. Evans, W.D., On non-integrable square solutions of a fourth order differential equation and the limit-2 classification, J. London Math. Soc. (2) 7 (1973) 343-354. Everitt, W.N., Some positive definite differential operators. J. London Math. Soc. 43 (1968) 465-473. Everitt, W.N., On the limit-point classification of fourth order differential equations, J. London Math. Soc. 44 (1969) 273-281. Hinton, D.B., Limit-point criteria for differential equations, Canad. J. Math. 24 (1972) 293-305. Hinton, D.B., Limit-point criteria for positive definite fourth order differential operators, Quart. J. Math. 24 (1973) 367-376. Hinton, D.B., On the eigenfunction expansions of singular ordinary differential equations, J. Differential Equations 24 (1977) 282-308.
[12]
Kauffman, R.M., On the limit-n classification of ordinary differential equations with positive coefficients, Proc. London Math. Soc. (3) 35 (1977) 496-526.
[13]
Kauffman, R.M., The number of Dirichlet solutions to a class of linear ordinary differential equations, J. Differential Equations 31 (1979) 117-129. Robinette, J., On the Dirichlet index of singular differential operators, in preparation
[14] [15]
Walker, P.W., Deficiency indices of fourth order singular differential operators, J. Differential Equations 9 (1971) 133-140.
This page intentionally left blank
This Page Intentionally Left Blank
Spectral Theory of Differential Operators I. W. Knowles and R. T. Lewis (eds.) © North-Holland Publishing Company, 1981
SPECTRAL AND SCATTERING THEORY FOR PROPAGATIVE SYSTEMS Martin Schechter Yeshiva University New York, New York
We discuss a system of equations that describes many (if not most) wave propagation phenomena of classical physics. We consider spectral and scattering theory under minimal assumptions on the coefficients. 1.
INTRODUCTION.
Many wave propagation phenomena of classical physics are governed by systems of partial differential equations of the form (1.1)
E( x )
n
<JU
\'
-;:-It =
,
j
L
=1
A ~ _ -iAu j
dX
j
where x = (xl"" ,x ) E lRn, u(x,t) is a column vector of length m describing the n state of the medium at position x and time t (cf. Wilcox[27]). Here E(x) and the Aj are real, symmetric m x m matrices with the following properties: (a) (b)
E(x) is a uniformly positive definite function of x; the A. are constant. J
From the point of view of spectral and scattering theory it is desirable that solutions of (1.1) be of the form u
=
e- itH u0'
where H is a selfadjoint operator. This would require that H be an extension of E-1A. vihen E = 1, one can easily obtain a selfadjoint realization Ho of A in H = [L2]m using Fourier transforms. On the other hand, if E f 1, the operator E-1A need not be Hermitian on H. However, it is Hermitian on the Hilbert space Hl with scalar product (1 .2)
(u , v) 1 =
f
v (x) * E(x) u (x)
dx .
(The asterisk denotes the conjugate transpose.) When E(x) is uniformly bounded, one can show that the operator E-1H is selfadjoint on Hl (cf. [27]). However, o when E(x) is unbounded, this operator need not be selfadjoint. It is not even clear that E-1A has an extension that is selfadjoint. 361
362
MARTIN SCHECHTER
We shall study the system (1.1) without assuming E(x) bounded and assuming very little more than (a) and (b). This leads to some difficult technical problems. It is surprising that one can obtain any results at all. However, we show that one can get results for such systems comparable to those known for systems obeying more stringent conditions. Our next step is to add the usual stronger hypotheses to A but still allowing E(x) to be unbounded. We are then able to improve the results. The more we assume about A, the stronger the results become. However, at no point are we required to assume E(xl bounded. 2.
THE EXTENSION.
Our first difficulty is on H . It is not even clear l one, we do not know how many with the problem of deciding is given by Theorem 2.1.
due to the fact that E-1H o need not be selfadjoint that it has a sel fadjoint extension. Even if it has it has. If it has more than one, we are confronted which one to choose. Our solution to these problems
There exists an extension H of E-1A determined uniquely by f and A.
If there are constants a, C such that
J
(2.1)
IE (x) Idx
~ C(1 + Ix I )a
I x-y 1<1 then H is selfadjoint.
We construct H as follows.
Put h
A(s) = -
(2.2) and let
u denote
(2.4)
(Au) E
J J
the Fourier transform of u.
(2.3) If u, f
I A· s ·,
j=l
Hl • we shall say that u '*
E
=
Then for any test function u
A(s) u(s).
D(H) and Hu
J v(s) A(s) u(s) ds A
=
=
f if and only if
(f,v)l'
Clearly H is well defined and Hermitian on H . It depends uniquely on E and A, l and it is an extension of E-1A. If (2.1) holds, then H is selfadjoint. Now that we have constructed H, we turn to the study of its spectrum. well known that atHol = (-00,00)
(2.5)
=
lR.
Under mild conditions the same is true for H. Put F(x) = E(x)1/2 and let denote the (orthogonal) projection onto N(Hor- in H. We have Theorem 2.2. If D(H~) H for some k, then
C
It is
P o
Hl and (F - 1 )(Ho - i)-k Po is a compact operator on
363
SJ'LCl'RAL .ISO SCA1IU<JNC THI:CJJil' FOR l'ROP.4G.l'J'JI 'L S,'STUJS
o(H)
(2.7)
=
lR.
Another way of stating the hypothesis of Theorem 2.2 is that (F - 1 )P o is Ho-compact. Sufficient conditions for this to hold are given in [16J. k
Next we turn to scattering theory. Let J be a bounded linear map from H to We shall say that u E D(W±(Ho,H,J)) and W±(Ho,H,J)u = f if
Hl . (2.8)
Fi rst we have Theorem 2.3.
Asswne that "
(Ju)
(2.9)
p sueh that
a > 1, 2 ~ P ~
00
and
J
(1 + Ixl)a (
(2.10)
J(C) u(E,),
IF(Y) - 112 dy)1/2
LP .
E
I x-y 1,1 Then
(2.11)
Theorem 2.4.
Let J
con stan t matrix J some k
>
O.
=
J(x) be
sv,(,h
ok o
If D(H )
C
t
of"
h1t J H
00
C
00
X,
i/rzd
SU[~)'OSI::.~
twd (J - J 0) (i - Ho
rk
th:,:t thel'e is
:1
Po is ,'omrue [, j',)£'
Hl and (2.10) haZels, then
(2.12) 3.
HJ
N(H ).L o SYSTEMS WITH CONSTANT DEFICIT.
C
D(W (H ,H,J)) ±
0
Now we shall strengthen our hypotheses on A. This will allow us to weaken the assumptions on E. We shall say that the system (1.1) has (!Oilstant defi,!it if the matrix (2.2) has constant rank for 0 t- E, E lRn. Let S be the selfadjoint realization of (1 - £\)1/2 in L2, where II is the Laplacian in lRn. We can strengthen Theorem 2.2 in this case to read Theorem 3.1.
Ass['<;ne that the suskm (1.1) has constant deficit.
anel (F - 1 )S-k Po is a ,'ompaet orcmtoY' on H for;
Corollary 3.2.
Let (1. 1) have
is bounded and tends to 0 as
c,)lwLcm(
J Ix-y I<1 Ix I 00, -7
SC"rIe
dej'ieil..
Tf D(Sk)
C
Hl
k, then (2.7) hui-ds.
IF
IF (y) - 1 12 dy then (2.7) holds.
Corollary 3.2 is obtained from Theorem 3.1 by finding a sufficient condition for (F - 1) to be sk-compact (cf. [16J). We can also improve Theorem 2.4 in this case.
We have
364
MARTIN SCHECHTER
Theorem 3.3.
Assume that (1.1) has constant deficit and that
J
IF(y) 12 dy
Ix-y I<1 is bounded.
Let J = J(x) be a matrix function of x> and suppose there is a con-
stant matrix J o such that JoHo c HoJ o and
J
IF(y) (J(y) - J ) 12 dy
Ix-yl
4.
0 -+
00.
Then (2.10) implies (2.12,).
UNIFORMLY PROPAGATIVE SYSTEMS.
One can strengthen these theorems even further if one is willing to assume more about the operator A. The system (1.1) is called uniformly propagative if the roots A of the equation det(Al - A(!;))
(4.1)
=
0
have constant multiplicities and constant algebraic signs for 0 f such systems we can state the following Theorem 4.1. form (2.9).
ERn.
For
Assume that (1.1) is unifDl'mly propagative and that J is of the Assume further that there are constants a> P such that
(4.2)
2 2 P2
and (2.10) holds.
00,
a >
1
-
n - 1 p-
Then (2.11) holds.
Note that (4.2) allows Theorem 4.2.
i;
a
to be
negative when n
>
3.
We also have
Assume that (1.1) is uniformly propagative and ozl of the hypotheses
of Theorem 3.3 hold. Assume further thot thel"e ope constants a, p such that (4.2) and (2.10) hold. Then (2.12) holds.
5.
ASYMPTOTIC COMPLETENESS.
Until now we have not discussed the question of completeness. We shall show that such results can be obtained as well without assuming E(x) bounded. We shall call the wave operators W+(H ,H,J) complete if their ranges are dense in Hac (H), 0 the subspace of absolute continuity of H (cf. [12]). We have Theorem 5.1. Assume that (1.1) is uniformly propagative and that the hypotheses of Theorem 2.3 hold with p
=
00.
Then the lJave operators are complete.
Theorem 5.2. Let all of the hypotheses of Theorem 4.2 hold lJith P /Jave operators are complete. We can do a bit better if we assume more about A. is the set of those i; E Rn which satisfy det (1 - A(t;))
O.
=
00.
Then the
The sloZJness sUI'face of A
SPLC1'K4L AND SC47TLl
365
If Al (1;), ... ,Am(i;) are the roots of (4.1), then the slowness surface of A is the n {i; E lR I Ak(d = 11. We shall call the system (1.1) convex
union of the sheets
if the sheets of the slowness surface are all convex.
Theorem 5.3.
We have
Let (1.1) be conve", UniroY'mly l'ropagative, and let all or the
hypotheses ur Theorem 4.1 be satisj"l:ed ".lith (4.2) replaced by
1 .:: p .:: ro,
(5. 1 )
((
>
1 - (n
~n 1 ) p'
((
>
O.
Then (2.11) holds, and the wave o!!er'otol'S .]re eomplete.
Theorem 5.4.
(1.1) convex
FOf'
Theorem 3.3 be satisfied. such that (2.10) holds.
6.
I'ropagative let the hypotheses or
Assume [hal; ther'e are eonstants
0"
p satisfying (5.1)
Then (2.12) holels, anel the Wave oper'ators ar'e complete.
COMMENTS CONCERNING THE PROOFS. Propagative systems (1.1) in which E(x) is unbounded provide excellent exam-
ples of two-space spectral and scattering theory in which the two Hilbert spaces are not equivalent. in the theories.
This situation presents difficult but interesting problems
For our completeness results we have appl ied the theory in [20J.
In 1976 the author [18, 19J formulated a new criterion for the existence of wave operators which generalized the well known criterion of Cook [5J. generalized by him to two space scattering [21, 22J.
This was later
These results were subse-
quently simplified and generalized by Simon [25J, Kato [13J, Davies [6J, Enss [lOJ, Ginibre [llJ, and Combes-Weder [4J. of the others.
However, no one of these results includes all
This leads to the natural question whether there exists a single
theorem that implies all of the rest.
This can be answered in the affirmative.
The following theorem implies all of the general izations of Cook's theorem found in the references mentioned above. Theorem 6.1.
Let Ho' H be selfadJoint opCl'ators on Hilbert spaces H ' H, respeco tively,and let J be a bounded linear' operator· from Ho to H. Put
u ot
= e- itHo
u,
v t
= e- itH
v,
W(t)
= e itH
Ju
ot
·
Let U be an element of H ' and assume that there are complex valued functions o f(\), g(\), a peal number a and a function ~(t) from the interval [a,oo) to H such that (1 )
O(g(H) * ) is dense in
(2)
u E D(Ho ) n D(f(Hol}
(3)
W(a)u
(4)
For each t ~ a and V E D(Hg(H) * )
E
H
D(g(H))
(Ju t,H9(H) * v) - (JH u t,g(H) * v)
o
(5)
The function
0 0
cj> (t)
(cj>( t)
satisries
r
a
11¢(t)1I dt
<
00
, v)
366
,II lRTTN SCJlE!CHTI:'R
then the fol (a)
(b) (c)
7.
cone l/Asions ho ld:
g(H)W(t)u conver;lcs to Bomc elr!?lent h in H 1im supIlW(t)f(H )u - hll < 1 im supll [g(H)J - Jf(Ho)Ju til t-+oo 0 t-+oo 0 lim supll[W(s) - W(t)Jf(Ho)u l < 2 lim supll[g(H)J - Jf(H )Ju til s,t-7oo t+oo 0 0
REMARKS.
Spectral and scattering theory for uniformly propagative systems were studied by Wilcox [27,28,29J. He proved existence of the wave operators (i.e. ,(2.12)) under the assumption (7.1) E(x) - 1 = O( lxi-a) as Ixl -+ for some (1 > 1. Completeness was proved by Mochizuki [14J, Birman [2J, Oeic [7], Suzuki [26J, Yajima [30J under assumption (7.1) and various other assumptions. It was proved by Schulenberger-Wilcox [23], Birman [2], Oeic [8] and Schulenberger [24J under the assumption m
f (1 + Ix I ) S IE (x) - 1 12 dx < (7.2) for some B > n together with various other stipulations. Deift [9J was able to remove the other assumptions. Schechter [17J proved completeness under the assumption (2.10) with p = 00, a > 1. This includes all of the other results. In all of these results it is assumed that E(x) is bounded and J is the identity operator Ju = u. (When E(x) is bounded the Hilbert spaces Hand Hl consist of the same functions. In this case we can take J as the identity operator. If E(x) is unbounded, we cannot use the identity operator for J.) The author's paper [21J was the first to allow E(x) to be unbounded. The present paper shows that no generality is sacrificed; all of our results are stronger than those mentioned. For systems that are not uniformly propagative very little work has hitherto been done. Avila [lJ proved the existence of the wave operators under condition (7.2) with S = 4 in addition to (a), (b) and the boundedness of E(x). His result is generalized by our Theorem 2.3. Nenciu [15J has considered eigenfunction expansions under the conditions that (1.1) has constant deficit, E(x) - 1 is bounded and dies down exponentially at infinity, and (a), (b) hold. Proofs of the results announced in the present paper will be published elsewhere. 00
REFERENCES [lJ
Avila, G. S. S., Spectral resolution of differential operators associated with symmetric hyperbolic systems, Applicable Analysis 1 (1972) 283-299.
[2J
Birman, M. S., Some applications of a local criterion for the existence of wave operators, Ookl. Akad. Nauk SSSR 185 (1969) 735-738 (Russian).
[3J
Birman, M. S., Scattering problems for differential operators with perturbation of the space, Izv. Akad. Nauk SSSR 35 (1971) 440-455 (Russian).
Sl'ECTRAL AND scnTUUNC rIflJJRY FOR l'ROl'ACTI1!T SYSn,A[S
367
[4J
Combes, J. t4. and Weder, R. A., New criterium for existence and completeness of wave operators and applications to scattering by unbounded obstacles, to appea r.
[5J
Cook, J. M., Convergence of the 82-87.
[6J
Davies, E. B., On Enss' approach to scattering theory, Duke Math. J. 47 (1980) 171-185.
[7J
Deic, V. G., The local stationary method in the theory of scattering with two spaces, Dokl. Akad. Nauk SSSR 197 (1971) 1247-1250 (Russian).
[8J
Deic, V. G., Application of the method of nuclear perturbations in two space scattering theory, Izv. Vyss. Ucebn. Zared (1971) 33-42.
[9]
Deift, P., Classical scattering theory with a trace condition, Thesis, Princeton University (1976).
M~ller
wave matrix, J. Math. Phys. 36 (1957)
[10] Enss, V., Scattering theory of Schrodinger operators, in : Velo, G. and Wightman, A. S. (editors), Rigorous Atomic and Molecular Physics (Plenum, New York, 1980/81). [11] Ginibre, J., La methode "dependant du temps" dans le probleme de la completude asymptotique, to appear. [12J Kato, T., Perturbation Theory for Linear Operators (Springer, New York,1966). [13] Kato, T., On the Cook-Kuroda criterion in scattering theory, Commun, Math. Phys. 67 (1979) 85-90. [14J Mochizuki, K., Spectral and scattering theory for symmetric hyperbolic systems in an exterior domain, Pub1. RIMS, Kyoto Univ. 5 (1969) 219-258. [15] Nenciu, G., Eigenfunction expansions for wave propagation problems in classical physics, Com. Stat. Pen. En. Nuc., Inst. Fiz Atom., Bucharest, FT-113-1975. [16] Schechter, M., Spectra of Partial Differential Operators (North-Holland, Amsterdam, 1971). [17J Schechter, M., A unified approach to scattering, J. Math. Pures Appl. 53 (1974) 373-396. [18] Schechter, M., The existence of wave operators in scattering theory, Bull. Amer. Math. Soc. 83 (1977) 381-383. [19] Schechter, M., A new criterion for scattering theory, Duke Math. J. 44 (1977) 863-862. [20] Schechter, M., Completeness of wave operators in two Hilbert spaces, Ann. Inst. Henri Poincare, 30 (1979) 109-127. [21] Schechter, M., Scattering in two Hilbert spaces, J. London Math. Soc. 19 (1979) 175-186. [22] Schechter, M., Wave operators for pairs of spaces and the Klein-Gordon equations, Aequationes Math. 20 (1980) 38-50.
368
MARTEN SCHECHTER
[23] Schulenberger, J. R. and Wilcox, C. H., Completeness of the wave operators for perturbations of uniformly propagative systems, J. Func. Anal. 7 (1971) 447-474. [24]
Schulenberger, J. R., A local compactness theorem for wave propagation problems of classical physics, Ind. Univ. Math. J. 22 (1972) 429-432.
[25] Simon, B., Scattering theory and quadratic forms: Commun. Math. Phys. 53 (1977) 151-153.
On a theorem of Schechter,
[26] Suzuki, T., Scattering theory for a certain non-selfadjoint operator, t~emoirs Fac. Liberal Arts and Ed. 23 (1974) 14-18. [27] Wilcox, C. H., Wave operators and asymptotic solutions of wave'propaqation problems in classical physics, Arch. Rat. Mech. Anal. 22 (1966) 37-78. [28] Wilcox, C. H., Steady-state wave propagation in homogeneous anisotropic media, Arch. Rat. Mech. Anal. 25 (1967) 201-242. [29] Wilcox, C. H., Transient wave propagation in homogeneous anisotropic media, Arch. Rat. Mech. Anal. 37 (1970) 323-343. [30] Yajima, K., Eigenfunction expansions associated with uniformly propagative systems and their applications to scattering theory, J. Fac. Science, Tokyo Univ. 22 (1975) 121-151.
Spectral Theory of Differential Operators I.W. Knowles and R. T. Lewis leds.) © North-Holland Publishing Company, 1981
SPECTRAL ANALYSIS OF MULTIPARTICLE SCHRODINGER OPERATORS .
1
Barry S1mon Department of Mathematics California Institute of Technology The first lecture is an introduction to some recent work by Peter Perry, Israel Sigal and me [2,3J on the spectral analysis of N-body Schrodinger operators. Our work is based in part on some beautiful ideas of Eric Mourre [lJ. Given masses m. and functions (potentials) on RV, V.. , with 1 < i < j < N, J 2 (N- 1 ) 1J (N- 1) we define an operator H on L (R V ), as follows: think of RV as N tuples N V of vectors rj in R with I mjrj = O. Let V = . I . V1'J' (r.1 - r.) J and let Ho be the 1
1 <J
Laplace Beltrami operator associated to the metric
I
2
mjdr j .
Then H = Ho + V.
Perry, Sigal and Simon consider potentials V.. = V~~) + V(~) + VP) where 1J 1J 1J 1J 2 (1) the following six operators are -~-compact on L2 ( R\J,I: (1) (l+lxl)V ; (4) v(3); (5) (1+lxl)IJV(3); (2) (1+lxl)v(2\ (3) (1+lxl)2 vv (2); (6) (1+[xl)21JIJV (3). Roughly speaking any x- 2-E: potential is allowed; slower falloff requires more smoothness but very slow falloff (e.g. (£nr) -1-£-) is allowed. Theorem [2,3] Under the above conditions: (i) H has empty singular continuous spectrum. (ii) The thresholds of H are a closed countable set. (iii) Non-threshold eigenvalues are of finite multiplicity and such eigenvalues can only accumulate at thresholds. References [1 J P. Perry, 1. Sigal and B. Simon, Bull. Am. ~1ath. Soc. 1(1980) 1019. [2J P. Perry, I. Sigal and B. Simon, Ann. Math. to appear. [3J E. Mourre, Commun. Math. Phys. 78(1981) 391.
SCHRODINGER OPERATORS WITH ALMOST PERIODIC POTENTIALS In the second lecture some general conjectures and results about operators of the form -d/dx 2 + V(x) = H 2 on L (_oo,oo), where V is a (Bohr) almost periodic function,are discussed. This is a subject of intense current interest [1,2,4,5,9J. Earlier significant results can be found in [3,6,7,8J. 369
370
Il. SlMON
Two main features are to be expected: (i) The spectrum of H is a Cantor set for "most" almost periodic V. (ii) If V is multiplied by a sufficiently large constant, H will have dense point spectrum at low energies. Connected with (i) is anomalous long time behavior for the quantity [lJ. So far the proven results concerning (i) and (ii) are somewhat limited: (i) is proven for generic 1 imit periodic V [1,5J, and (ii) has been announced [2J for some special finite difference analogs of H. Sarnak [9J has proven (ii) for such operators with V a special complex valued function. (, exp(-itH)
One interesting application is to think of H as a Hill operator (linear stability operator in classical mechanics) as would arise in the study of the ri ngs of Saturn. [1]. References [lJ [2J [3J [4J [5J [6J [7J [8] [9J
J. Avron and B. Simon, California Tnstitute of Technology Preprints. J. Bellissard and D. Testard, CNRS - Marseille Preprints.
E. Dinaburg and Va. Sinai, Funk. Anal. i. Pril ~(4), (1975) ,8. R. Johnson, U. S. C. Preprints. J. Moser, ETH Preprints. H. Russman, Proc. N. Y. Academy Sci. to appear. R. Sacker and G. Sell, J. Diff. Eqn. 12(1974), 429; 22(1976), 478; 22(1976),497. M. Shubin, Russian Math. Surveys, 11(1978), #2, 1. P. Sarnak, Courant Preprint.
Sherman Fairchild Scholar, on leave from Princeton University.
Spectral Theorv of Differential Operators I.W. Knowles and R. T. Lewis (eds.) © North-Hal/and Publishing Company. 1981
ESTIMATES FOR EIGENVALUES OF THE LAPLACIAN ON COMPACT RIEMANNIAN MANIFOLDS Udo Simon Fachbereich Mathematik Technische Universitat Berlin o - 1 Berlin 12 FRG
Let (M,g) be a closed, connected Riemannian manifold of dimension n ~ 2 with sectional curvature K. Let v denote the corresponding covariant differentiation, R the scalar curvature (normed such that R = 1 on the unit sphere), dw the volume element and Ko: = min K, Kl : = max K on (M,g); let Hess(f) denote the Hessian and ~f = trace Hess(f) the Laplacian of f E Coo(M). ~ is an eigenvalue of ~ if ~f + Af = 0 has a nontrivial solution. In local coordinates (u i ) on M, g .. and gij denote the components of the 1J
tensors 9 and g-l , respectively; raising and lowerlng of indices are defined as usual; the components of the Riemannian curvature tensor and the Ricci tensor Ric are denoted by R~'k and R.. respectively; the components of the higher order 1J 1.1 covariant derivatives of f E COO(M) are denoted by fi' f ij , f ijk , etc. The interest in estimates of eigenvalues of the Laplacian on Riemannian manifolds comes from the following fact: one knows that the geometry of a closed Riemannian manifold determines the spectrum {O = ~o ~ Al ~ A2 _ ... } which lies on the nonnegative real half line and is purely discrete, tending to infinity; all eigenvalues A are of finite multiplicity, i.e., the corresponding eigenspaces EA have finite dimension. But there are only a few examples of manifolds where the spectrum is completely known, and even for such a simple surface as an ellipsoid one does not know the exact value of Al' This explains why one tries to compute estimates for eigenvalues by geometric data (for an introductory survey cf. [9J). In [2J we have given the following estimate for pinched (i .e., KO = oKl , 0 < 6 ~ 1) 2-manifolds. Theorem A. Let M be closed, n = 2, 8 -> 1 Then: 3 (i) There are exactly 3 eigenval ues Al ~ '\2 ~ A3 in the interval [2K O,2K l J counted with multiplicity) and '\4 ~ 6K O ~ 2K l · (ii) EqualitY'\l = 2KO or'\3 = 2Kl or'\4 = 6K O implies that (M,g) is isometrically diffeomorphic to the standard sphere S2(K). The techniques for the proof which we developed in [2J come from the fact that eigenfunctions and eigenvalues on standard spheres can be characterized by certain systems of partial differential equations (cf. [7J, [lOJ, [5J); the 371
372
UDO SIMON
pinching of the sphere suggests that we define related expressions. In the following we are going to demonstrate that in higher dimensions(n::.3) our method works on a large class of spaces, which in particular contains many homogeneous spaces (cf. [4J, [6J). 1. Definition. Let (M,g) be connected; we call the Ricci tensor of (M,g) cyclic if VkR ij + ViR jk + VjR ki = O. The following lemma is the basic tool for an integral inequality. 2.
Let (M,g) be connected, n ~ 2, f E Coo(M).
Lemma. [8J.
l:iIlIIHess(f)112 = 2
L
Then
Kij(si - Sj)2 + (Hess(f),Hess(M)) + IlvHess(f)1I 2
i <j
f
ij k
f {2V Rjk - VkR ij }, i where sl,···,sn are the eigenvalues of Hess(f), El,···,E n are corresponding orthonormal eigenvectors in the tangent space and Kij = K(Ei,E j ) is the sectional curvature defined by span (E.,E.).,..; ( ,) denotes the inner product on correspond1 J 1TJ ing tensor spaces, induced by the metric of (M,g). +
3. Computations. We use (2) to prove an integral inequality for fixed f E EA' As the nodal set for f E EA is nowhere dense [3], the zeroes of 1 grad(f)1I 2 are nowhere dense and therefore there exists a uniquely determined continuous function R* on M such that Rijf.f. = (n - 1)R*lIgrad(f)11 2 . J
1
Denote the eigenvalues of the Ricci tensor by r l ~ r 2 ~ ... ~ rn and let (n - l)r: = min r l (p), (n - 1 )r*: = max rn(p), so KO = r = R*(p) = r* = Kl for p E M. Define the symmetric (3.0) tensor B(f) by (3.a) B(f)ijk:
=
f ijk +
~(\+ 2R*)gi/k
+
~(\ - nR*)(gikfj + gjkfi);
then (cf. [2,(3.3)J in the special case of an Einstein space) (3.b) fIIB(f)1I2 jllvHess(f)1I 2dw - ~(3A2 - 4(n - 1 )AR* + 2(n - 1 )nR*2)G, where G: = jllgrad(f)1I 2 dw. (3.c) For the following computations we assume KO > 0 on (M,g), which implies that the Ricci tensor is positive definite; but most of the computations are true without this assumption. From [2, Lemma 1 .2J we get for f (3.d)
E\
2 (A - (n -1 )r*)G ~ jIlHess(f)11 dlu ~ (A - (n -1 )r)G.
Analogously [2, (1.3)J implies for f (3.e)
E
2 j
L K.. (S.
i <j
1J
-
1
For a cyclic Ricci tensor we have
E
EA
s.)2 dw J
>
-
2(n - l)KO(A - nr*)G.
liS1111HTnS H)R UCL\'J'\U,'ES OF THn LAPLACUN
373
where we use the Ricci identity to prove the right hand side inequality. Integration of (2) and the computations in (3) imply the following integral inequality. 4. Lemma. Let (M,g) be closed and connected, n ~ 3, with positive sectional curvature and cyclic Ricci tensor. Then for f E EA (4. 1 ) 0 ~ JI B(f) I 2 dw + P(A) G, where PtA) is the following polynomial of second order l-n 2 4(n+3) n-l 2 (4.2) PtA): = n+2 A +(n-l)(2K +5r - (n + 2) r*)A+2n n+2(r - (n+2)K r*) O o + 4 (n - 1 ) 2 r* (r - r*). Under suitable curvature conditions (e.g. pinching conditions) the polynomial PtA) has two real zeroes A(1), A(2) which depend on the geometric data n = dim M and the curvature bounds K ' r, r*. As PtA) > 0 for A E (A (1) ,A (2)), the integral O inequality implies immediately the following result. 5. Theorem. Let (M,g) be closed and connected with positive sectional curvature and n > 3. If the Ricci tensor is cyclic, there is no eigenvalue in the interval (A(l),~(2)). We were interested in demonstrating how to get integral inequalities; we restrict ourselves by giving explicit values for the above interval in the case of Einstein spaces; then A(1) = nR and 1.(2) = 2(n + 2)K - 2R. The interval (A (1) ,A (2)) is nonempty only if 2KO ~ R, or 6 ~ ~ ,respectively. In [2] we proved that (M,g) must be a Riemannian sphere if an eigenvalue A fulfills A = A(l) orA=A(2). Similar computations can be made for conformally flat spaces (n ~ 3); then L: = Ric - y,Rg fulfills Codazzi conditions, so estimates similar to (3.f) can be given. D. Barthel and R. Kumritz [1] have shown that our technique works also when one considers the related situation for the Laplacian plus a potential on Einstein spaces (because of a result of Cheng [3] their assumption on the nodal set of the eigenfunction is superfl uous (cf. [9], §5)).
[1]
REFERENCES .. Barthel, D. and Kumritz, K., Laplacian with a potential. Proceedings Colloquium Global Analysis - Global Differential Geometry. TU Berlin 1979. Lecture Notes Mathematics 838. Springer, 1981.
[2]
Benko, K., Kothe, M., Semmler, K.-D. and Simon, U., Eigenvalues of the Laplacian and curvature. Colloquium Math. 42, 19-31 (1978).
[3]
Cheng, S.-Y., Eigenfunctions and nodal sets. (1976) .
Comm. math. Helv. 51,43-55
374
UDO SIMON
[4J
D'Atri, J. E. and Ziller, W., Naturally reductive metrics and Einstein metrics on compact Lie groups. Memoirs AMS 18, No. 215 (1979).
[5J
Ga1lot, S., Varietes dont le spectre ressemble a celui de la sphere. Comptes Rendus Acad. Sci. Paris 238, 647-650 (1976).
[6J
Gray, A., Einstein-like manifolds which are not Einstein. Dedicata 7, 259-280 (1978).
[7J
Obata, M., Certain conditions for a Riemannian manifold to be isometric with a sphere. J. Math. Soc. Japan 14, 333-340 (1962).
[8J
Simon, U., Isometries with spheres.
[9J
Simon, U. and Wissner, H., Geometry of the Laplace operator. Kuwait Conference on Algebra and Geometry, Feb. 1981. Proceedings. To appear.
Geometriae
Math. Zeitschrift 153, 23-27 (1977).
[lOJ Tanno, S., Some differential equations on Riemannian manifolds, J. Math. Soc. Japan 30, 509-531 (1978).
Spectral Theory of Differential Operators I.W. Knowles and R. T. Lewis leds.) © North·Holland Publishing Company, 1981
THE SQUARE INTEGRABLE SPAN OF LOCALLY SQUARE INTEGRABLE FUNCTIONS Philip W. Walker Department of Mathematics University of Houston Houston, Texas 77084 U.S.A.
A method using limits of inverses of Gram type matrices is given for determining the square integrable span of a finite sequence of locally square integrable functions. Whenever S is a Lebesgue measurable subset of the real numbers L2 (S) will denote the Hilbert space of equivalence classes of complex valued functions y defined on S with the property that (
J
lyl2
s
exists and is finite. We suppose that each of E and Ek for k = 1, 2, able subset of the real numbers, that for k = 1, 2, . . . ,
is a Lebesgue measur-
and that U E
k=l
k
=
E
We take n to be a positive integer and each of Y1' . . . , Yn to be an equivalence class of Lebesgue measurable complex valued functions defined on E. We suppose that Yi is in L2(Ek) for i = 1, . . . , n and each positive integer k and that (Y1, Yn) is linearly independent over E1. hence over each Ek and over E . Let V be the spa n (i. e . the set of linear combinations) of (Y1' and let L be L2 (E) 11 V It is our purpose here to give a method for determining the space L. The principal difficulty lies in the fact that non-trivial linear combinations of function classes not in L2(E) may be in L2(E). For an example consider ([nJ, [Y2J) where Y1 (x) = 1 and Y2(x) = 1 + l/x for all x?: 1 . The problem of finding L occurs in the study of singular differential operators. See, fot example, [lJ and [2J Theorem XIII.3.8. Earlier results concerning the dimension of L may be found in [31 and [4J. (We caution the reader that in [4J "span" is used to denote the dimension of L.) Here we go further in that we give a constructive way of representing a spanning set for L For each Lebesgue measurable subset f = (f1, . . . , fQ) of members of matrix whose (i,j) entry is
S of the real numbers and each n-tuple L?(S) we will denote by G(S,f) the n
I
S
Thus
G(S,f)
~
f.f . . J 1
is the transpose of the Gram Matrix of 375
(f1, . . . , fn) .
x
n
376
PHILIP WALKER
G(S, f)
Obviously
is Hermitian and G(S, f)
(1 )
=J
f*f
S
bn ) t , where t where * denotes conjugate transpose. If b = (bl' notes transpose, is an n x 1 matrix of complex numbers and v = bl f 1 + bnf n then v = fb so v = b*f* and from (1) we have that (2) vv = b*G(S,f)b .
de. +
j[S
Since the left side of (2) is non-negative we see that G(S,f) is non-negative definite. If (fl, . . . , fn) is linearly independent over S and at least one bi is not zero then v F 0 so the left side of (2) is positive. Thus independence of (fl, . . . , fn) implies G(S,f) is positive definite hence non-singular. When this is the case G-l(S,f) will denote (G(S,f))-l. Our main result is given in the following theorem. THEOREM.
There is an
n
x
M such that
n matrix
lim G-1(Ek'(Yl' k-too
Moreover if
(u 1 '
, un)
i2. given Qy
(u 1 ' .
un) PROOF. G( E ,· ) k
un) = (Y 1 , is L
We will denote by G (.) •
Suppose that
by
by
y,
u, and
k j
~
k.
Then Gk(y) - Gj(Y)
G(E k \ Ej , y) ~ 0
=
so (3)
Hence from
Lemma
Thus
[5J
(See
Gj(Y) <: Gk(y) . following this proof we have that Gil(y) 2 Gzl(y) 2 • • . > 0 page 263.) there is an n x n matrix lim Gk1(y) = M .
M such that
k-too
Since each
Gk is Hermitian so is M. From Lemma 2 below we have that if j <: k then Gk1 (y) Gj(Y) Gk1 (y) <: Gk1 (y) Taking the limit as k 7 = with j fixed we may conclude that MG.(y)M <; M . J From (1), since M* M, we have that MG . (y) M = G. (yM) J J Since u yM and, as in (3), Gj(u) <: Gk(u) whenever j <: k we have shown that Gl(u) <: G2(u) <: • • • sM. Thus there is a matrix
B such that 1 im G. (u) j-too
J
B.
377
THE SQUARJ!-LY'J'/iC1HBLB SPAN
Therefore lim j-+w
i
u.u.
1 1
E. J
exists and is finite for i = 1, . . . , n . This shows that each ui is in LZ(E), and we may conclude that the span of (ul' . . . , un) is a subspace of
L.
If L consists only of the zero vector there is nothing more to prove. So suppose that L has dimension m where 1 s m s n. Let Z be the set of all n x 1 complex matrices (bl, . . . , bn)t such that blYl + . . . + bnYn is in L. Since (Yl' • . • , Yn) is linearly independent it follows from elementary algebra that the dimension of Z is also m. Suppose that • . • + bnYn
is a non-zero vector in d
=
1
vv
t
Note that
So
d
b*Gk(y)b
>
~
0 .
From (2) we hnve for
f
d as
Taking the limit as
k k
7
~
=
=.
tk From
vv
k
=
1, 2,
that
=
Lemma 3
below we have that
(b*b)Z < b*Gk(y)bb*Gi/ iy)b we may conclude that o < (b*b)Z/d < b*Mb ,
and this shows that Mb is not the zero n x 1 matrix. Since this is the case for each non-zero b in an m dimensional space it follows that M has rank at 1east m Since lUI, . . . , un) = (Yl"'" Yn)M and ( Y l " ' " Yn) is linearly independent it follows that the dimension of the span of (ul, . . un) is at least m Since we already have that the span of (ul, . . . , un) is a subspace of L and L has dimension m the proof is complete. LEMMA 1. If each of A and then 0 < B-1 s A-I .
B is a Hermitian
n
x
n matrix with
0
<
As B
PROOF. Let P be a non-singular matrix such that P*AP = I where is the n x n identity matrix and P*BP D where D is a diagonal matrix (See [5J Theorem 9.26.). Since 0 < A s B it follows that P*OP < P*AP s P*BP or o < I s D. Thus 0 < 1 s Di i for i = 1, • . . , n. Hence 0 < D~~ s 1 for i 1, . . . , n implying that 0 < D-l < I. From this it follows llthat o < p-l B-l(p*)-1 s p-lA-l(p*)-I, hence 0 < pp-lB-l(p*)-lp* s PP- 1A- l (P*)-l p* or o < B-1 s A-I . LEMMA 2. If each of B-IAB-l s Bthen
A and
PROOF. Since A s B we have the result follows.
B is a Hermitian
n
x
n ma tri x with
(B-l)*AB-l s (B- 1 )*BB- l .
But
(B- l )*
LEMMA 3. If A is an n x n positive definite Hermitian matrix and n x 1 matrix of complex numbers then (b*b)Z s b*Ab b*A- l b .
0
<
A <; B
B- 1 and
b is an
378
PHILIP WALKER
PROOF. Let Q be given by Q(x,y) = y*Ax whenever each of x and y is an n x 1 matrix of complex numbers. Since A is positive definite it is easy to verify that Q is an inner product. The Cauchy-Schwartz inequality for Q yields the fact that [y*Ax[ 2 ~ x*Axy*Ay . Let x = band y = A- 1 b . Then y* = b*A-1 and noting that b*b ~ 0 we have the desired result.
REFERENCES: LlJ
A. Devinatz, The deficiency index problem for ordinary self-adjoint differential operators, Bull. Amer. Math. Soc. 79(1973), 1109-1127.
[2J
N. Dunford and J. T. Schwartz, Linear operators, Part II, Interscience, New York and London (1963). ---
[3J
W. N. Everitt, Inegualities for Gram determinants, Quart. J. Math., Oxford (2), 8(1957), 191-196.
[4J
,Some ro erties of Gram matrices and determinants, Quart. J. Math. Oxford (2},9(1958 , 87-98:"" - --
[5J
D. T. Finkbeiner II, Introduction to matrices and linear transformations, 3rd edition, W. H. Freeman and Co.~San Francisco~.
[6J
F. Riesz and B. Sz.-Nagy, Functional Analysis, Ungar, New York (1955).
Spectral Theory of Differential Operators I. W. Knowles and R. T. Lewis (eds.) © North·Holland Publishing Company, 1981
ON A CONDITIONALLY CONVERGENT DIRICHLET INTEGRAL ASSOCIATED WITH A DIFFERENTIAL EXPRESSION S. D. Wray Department of Mathematics Mount Allison University Sackville, New Brunswick Canada
An inequality is established involving the Dirichlet integral of a second-order symmetric ordinary differential expression in a singular case. The inequality involves an integral containing the generalised Fourier transform of a function in the domain of the Dirichlet integral, integration being with respect to a spectral function of the differential expression. The proof of the inequality is based upon a corresponding equality in the regular case, which is established firstly.
1.
INTRODUCTION
We work in the interval [a,b), with - 00 < a < b s 00 , on the real line. As usual, let L[a,b), L 2 [a,b) and similar symbols denote the Lebesgue, complex integration spaces, AC absolute continuity and 'loc' a property to be satisfied on all compact sub-intervals of [a,b). A symbol such as '(fEE)' is to be read as 'for all f in the set E'. Consider the symmetric second-order ordinary differential expression by, for suitable functions f , M[f] ; w-l(_(pf')' + qf)
on [a,b)
(':= d/dx) ,
M
defined (1.1)
where the coefficients p,q and ware real-valued, Lebesgue measurable on [a,b) and satisfy the basic conditions -1 (i) p(x) > 0 (almost all xE[a,b» and p ELloc[a,b) (ii) qELloc[a,b); (1.2) (iii)
w(x) > 0 (almost all xE[a,b»
and wELloc[a,b). 2 L [a,b)
The theory takes place in the weighted Hilbert space tions f , defined on [a,b) and satisfying b
/wlfl
w
of complex func-
2 <00
a
the usual inner-product is used. Under the basic conditions (1.2), the linear differential equation M[f] ; At
on [a,b),
(1. 3)
where A is a complex parameter, is regular at all points of [a,b) in the sense of [7, Section 16.1], i.e. if tE[a,b) then the initial value problem defined by (1.3) and the conditions f(t); S , (pf')(t) ; n can be solved for arbitrary complex numbers sand n. It is assumed that b is a singular end-point in the sense of [7, Section 15.1], i.e. either b ; or b < 00 and at least one of p-l, q and w is not in L[a,b) .
379
380
STEPHEN D. WRIl Y
If cECa,b) and we consider the differential expression on the compact interval [a,c] then conditions (1.2) imply that both end-points a and c are regular in the sense of [7, Section 15.1]. If the Hilbert space L3[a,c] is defined in the same manner as L 2 [a,b) we may introduce the symmetric differential operator w TCc) as follows: 2 2 D(T(c)) {fELw[a,c]lf,pf'cAC[a,c], M[f]cL [a,c], w
f(a) cos a + (pf')(a)sin a fCc) cos S + (pf')(c)sin S
a
and
a}
(1. 4)
and T(c)f ; M[f]
(f"D(T(c))) ,
where a and S are real numbers in the interval [O,TI). It is well known that T(c) is self-adjoint in ~[a,c] and that its eigenvalues and eigenfunctions are as described below. Let ~(.,.) be the unique solution of (1.3) on [a,b) satisfying for all complex A the conditions ~(a,j); sin a and (p~')(a,A); -cos a. Then the eigenvalues of T(c) are countable, simple and are precisely the zeros of the entire function ~(c,·)cos S + (p~')(c, ·)sin S. They form an infinite sequence A
and has no finite cluster point. are given by
~n,c(x) ;
where
~n,c
The corresponding normalised
~o,c, ~l,c'···
1/2 rn,c ~(x,An,c)
(n 2 0, xc[a,c]),
r is in each case a positive normalisation constant chosen so that n,c 2 has unit norm in Lw[a,c].
Denote by 0c the right-continuous step-function on (_00,00) such that 0c(t) increases by r as t passes through A (n 2 0) , is otherwise constant, n,c n,c and is zero at a. It is well known that there is always a sequence of values of c , tending to b , for which 0c tends pointwise to a non-decreasing function 0 on (_00,00); we shall call this a spectral function of the differential equation (1.3). Throughout this paper, c will always tend to b through this sequence. By the spectrum of the differential equation (1.3) on [a,b) we shall mean the complement with respect to (_00,00) of the union of all open intervals over which 0 is constant; thus the spectrum is a closed subset of the real line. (This spectrum may be identified as that of a self-adjoint differential operator derived from M but we shall not need to introduce such an operator here.) Note that the spectrum depends on the choice of a and S in general. As usual, the differential expression M on [a,b), is said to be in the limitpoint (limit-circle) case at b if for each complex number A there is a (no) solution of (1.3) that is not contained in L~[a,b). We note here the well known facts that in the limit-point case at b, 0 tends pointwise to 0 as c tends continuously to b , and the spectrum §f (1.3) on [a,b) does not depend on
S. Now let L2 denote the Hilbert space of all complex-valued Lebesgue measurable functions F on (_00,00) such that
_{IFI 2 do
<
this being the Lebesgue-Stieltjes integral. We endow L2 with the usual innerproduct. As usual we may introduce a unitary transform from L~[a,b) onto L2 ,
381
ON "1 CO;\'DITIONALL \' CONVhRGhNT DIRICHLhT fNTJ£RAL
defined by
s
F (t)
and
lim s->b-
J f(x)¢(x,t)w(x)dx
(in L2 norm),
(1. 5)
a
FEL2 is its unitary transform. b
J wIf I a
2 =
Moreover,
2
00
J
(1. 6)
I F I dO' .
These results about a ,a and the unitary transformation are essentially to be found in [9]; the intr6duction of the coefficients p and w, and the use of finite b in some cases do not entail additional difficulties. The final definition we make is that of ~, which is the complex linear manifold of L 2 [a,b) consisting of all fEL 2 [a,b) such that ffAC [a,b) , pf'cL 2 ~,b\ 1 oc W W l DC f (a) = 0 if a = 0 and the limit
f
lim exists and is finite.
{plf' 12
+
(1. 7)
qlfl2}
Following Titchmarsh, we denote by ->b J {plf' 12 + qlfl2}
a
the limit in (1. 7)
2.
(fcE).
STATEMENT OF RESULTS
We state the main results of this paper in this section and give outlines of their proofs in the next section. Theorem 1. Let the differential expression M on [a,b) be defined by (1.1), let the coefficients p,q and w satisfy the basic conditions (1.2) and let the endpoint b be singular. Suppose additionally that the coefficients p, q and w and end-point b are so chosen that at least one of the following sets of conditions is satisfied (i) b O(w(x» as x->b ; , pEACloc[a,b) and p' (x) or
(ii)
b <
eo
and
d {(b _ x)-2 p (x)} dx
O(w(x» as x->b-
(2.1)
(xE[a,b»
(2.2)
or (iii) or
(iv) where hex)
x If there is a positive constant K such that q(x)
~
then we may replace w .Qy q in (2.1)
-Kw(x)
(i), (ii) or (iv).
Then the following inequality holds +b 2 2 ry {plf'l + qlfl } - If(a)l~cot a ~
J
eo £eo
2 tIF(t)1 dolt)
(fEE),
(2.3)
a
provided that the spectrum of the differential equation (1.3) is bounded below and M is either in the limit-point case at b, or in the limit-circle case with either S - 0 or S = n/2 , q = O. In (2.3), F is the unitary transform of f (see (1.5»
382
STEPHEN D. WRA Y
and the cotangent term is to be omitted if a Proof.
O.
This is given in Section 3 below.
Remark.
Theorem 1 significantly generalises earlier results in [10, p.200] and It is independent of the similar equality [5, Theorem 1]. See also the other references cited in [5]. Corollary. Let all the conditions of Theorem 1 hold. Then if ~ is the infimum of the spectrum we have
~785].
-+b
f
{plf'l
2
2
2
+ qlfl };:, If(a)1 cot a +
a
b
11
f
wlfl
2
.
(fEE)
a
Proof. This follows easily from the theorem on application of the Parseval formula in (1. 6) . Remark.
T2]:3.
See the similar result in [2, Theorem 4] and the others referenced in
PROOF OF THEOREM 1
A corresponding equality in the regular case is needed. Theorem 2. Let cE(a,b) and let p,q and w satisfy the basic conditions (1.2). Then the following identity holds 2 2 fC{plf' 12 + qlfl2} _ If(a) 1 cot a + If(c) 1 cot
S
a
L
.\
n=O
il,C
If
n,c
where f
n,c
fa
12
(fEE(c) )
(3.1)
c
wf1jJ
n,c
(n = 0,1,2, ... )
are the Sturm-Liouville coefficients of f on [a,cl. Proof. This follows the lines of the proo~of the corresponding identity in the singular case on [a,b); see [5]. The result is easily established for fED(T(c», and then it is extended to E(c), of which D(T(c» is a core in the sense of [6, p. 317]. Corollary.
f c {plf' I 2 !£
Under the conditions of Theorem 2 we have 2
+ qlfl }
2
2
If(a)1 cot a-
a n is a positive integer we have
2 If(c)1 cot S +.\
c
f
o,c a
wlfl
2
(fEE(c).(3.2)
(3.3) 2 for all fEE (c) orthogonal to 1jJ , .,. 1jJ 1 in L [a,cl 0, c n- ,C W (3.2) i f and only i f f A1jJo,c and in 0.3) i f and only is a complex constant. Proof.
).' f
There is equality in f = A1jJn,c' where A
This follows from the last theorem on application of the Parseval formula.
Remark. The inequality in (3.2) is proved by a different method in [1] by Amos and Everitt under the above minimal conditions on p, q and w, in the case
383
ON A CONDlTIONALL Y CONI'FRGENT DIRTCHLET INTEGR.4L
a =
B = n/2.
See also its review [3] by Beesack.
We now obtain Theorem 1 from Theorem 2 by means of a Tauberian argument which requires the following lemma. Lemma.
Let the spectrum of (1.3) be bounded below, with infimum
~.
Then
-A----~ ~ for all cE(a,b) and
o,c
A
lim
~
o,C
c-+b-
,
provided that either M is in the limit-point case at b or M circle case at b and also we have either B = or S = n/2
°
is in the limitand q = 0.
Proof. The proof of the similar result of Putnam in [8, pp.797-798] may be adapted to the present situation. We now proceed with the proof of Theorem 1. Let fEE be real-valued (the extension of the theorem from real-valued to complex-valued functions is straightforward), let b and let s > a. Define the function g by g(x) = { l - (x - a)/(s - a}f(x), (a" s), 0, (x> s), and apply Theorem 2 to g with c > s. If we express the series as a Stieltjes integral, we obtain 2 2 t /w(x)g(x)¢(X,t)dJ do Ct) = /{p(gl)2 + qg2} - f (a)cot a. (3.3) c _00 ~ ) a
X"
t
f
A calculation shows that the integral on the right is equal to 2 s 2 ~J~ + J ~\ (p(fl)2 + qf2) + ~l~ s-a as-a) s-a
S(l _~\f2 s-a) p,
(1 _
!
under the assumption that pEACloc[a,b). By the ReIly-Bray theorem, as c -; b = co T
I
T > max (O,~). we have
Let ~
f\a
J t Js w(x)g(X)¢(X,t)dX)2 do (t)
~
c
-+ /
~
If we make
proaches
s -+ b = F(t) in
00
t
fj
\~
S W
(X)g(X)¢(X,t)dX)2 do(t)
in the last integral, we find that the inner integral apL2 norm over [~,T] and hence that
JT t ~Js w(x)g(x)¢(x,t)dx)2 do(t) ~
+
JT tF 2(t)dO(t) ~
a
as s + 0 0 . If condition (i) of Theorem 1 holds, then we thus obtain the inequality of the theorem from (3.3) by replacing the limit w of integration by T, letting c -+ b, then s + b and then T -+ b. Clearly, one needs the lemma stated above. If, instead, condition (ii) of Theorem 1 holds, we proceed in a similar way, using g(x) = {I - (x - a)(b - s)/(s - a)(b- x)}f(x) , (a" x <; s), 0, (x> s). The theorem is proved in cases (iii) and (iv) by first changing independent variable from xE[a,b) to XE[O,B), where X(x) =
x -1 p ,
J a
B =
j
b
P-
1
(possibly 00),
a
and then using the case (i) and (ii) versions of the theorem on [O,B). The effects of this change of variable are described in sufficient detail in [4] to make it clear how to proceed here. The calculations are quite straightforward.
384
STEPHEN D. WRA Y
REFERENCES [1)
[2) [3) [4] [5J [6] [7J [S) (9) (10)
Amos, R. J. and Everitt, W. N., On integral inequalities associated with ordinary regular differential expressions, in: Differential equations and applications (Proc. Third Scheveningen Conf., 1977), North-Holland Math. Studies 31 (North-Holland, Amsterdam, 1975). Amos, R. J. and Everitt, W. N., On integral inequalities and compact embeddings associated with ordinary differential expressions, Arch. Rational Mech. Anal. 71 (1979) 15-40. Beesack, P. R., review of [1), Math. Reviews SOb:340l7. Everitt, W. N. and Halvorsen, S., On the asymptotic form of the TitchmarshWeyl m-coefficient, Applicable Anal. S (197S) 153-169. Everitt, W. N. and Wray, S. D., A singular spectral identity and inequality involving the Dirichlet integral of an ordinary differential expression, submitted for publication (November 19S0). Kato, T., Perturbation theory for linear operators, 2nd edn. (Springer, Berlin, 1976). Naimark, M. A., Linear differential operators, Part II (Ungar, New York City, 1965) . Putnam, C. R., An application of spectral theory to a singular calculus of variations problem, Amer. J. Math. 70 (194S) 7S0-S03. Sears, D. B., Integral transforms and eigenfunction theory, Quart. J. Math. Oxford (2) 5 (1954) 47-5S. Sears, D. B. and Wray, S. D., An inequality of C. R. Putnam involving a Dirichlet functional, Proc. Roy. Soc. Edin. Sect. A 75 (1975/76) 199-207.
