Spectral Representations for Schrödinger Operators with Long-Range Potentials (Lecture Notes in Mathematics)

Lecture Notes in Mathematics Edited by A. Oold and B. Eckmann 727 Yoshimi Saito Spectral Representations for Schrodin...

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Lecture Notes in Mathematics Edited by A. Oold and B. Eckmann

727 Yoshimi Saito

Spectral Representations for Schrodinger Operators With Long-Range Potentials r--:: - I ~

-

.

~

!>J;J.A~ I '~'.. ~~> ,

'~o£'~

,"~l?f'J L~_

Springer-Verlag Berlin HeidelberQ New York 1979

Author Yoshimi Saito Department of Mathematics Osaka City University Sugimoto-cho. Sumiyoshi-ku Osaka/Japan

AMS Subject Classifications (1970): 35J 10,35 P 25,47 A40 ISBN 3-540-09514 -4 Springer-Yerlag Berlin Heidelberg New York ISBN 0-387-09514-4 Springer-Verlag New York Heidelberg Berlirl Library of Congress Cataloglrlg In Publication Data Saito, Yoshlml. Spoctral representations for Schrodinger operators with long-range potentials, (Lecture notes In mathematics: 727) Bibliography: p. Includes Index, I. Differential equations. Elliptic, 2, Schrodlngor operator. 3. Scattering (Mathematics) 4. Spectral theory (Mathematics) I. Title. II. Senes: Lecture notos In mathematics (Berlin): 727, OA3.L28 no. 727 [OA377J 510'.88 [515'.353J 79-15958

This work IS subject to copyright. All rights arc reserved. whether the whole or part of the material is concerned, specifically those of trans lation, reprinting. re-use of illustrations, broadcasting. reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private usc. a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher, ~;:

by Springer-Verlag Berlin Heidelberg 1979 Printed in Germany

Printing and binding: Boltz Offsetdruck, Hemsbach/Bergstr, 2141/3140-543210

PREFACE

The present lecture notes are based on the lectures given at the University of Utah during the fall quarter of 1978.

The main purpose of the lectures was to present a

complete and self-contained exposition of the spectral representation theory for Schrodinger operators with longrange potentials. I would like to thank Professor Calvin H. Wilcox of the University of Utah for not only the opportunity to present these lectures but also his unceasing encouragement and helpful suggestions.

I would like to thank Professor

II

Willi Jager of the University of Heidelberg, too, for useful discussions during the lectures.

Yoshimi Saito

TABLE OF CONTENTS I NTRODUCTI ON

1

CHAPTER I Limiting Absorption Principle §l §2 §3 §4

Preliminaries Main Theorem Uniqueness of the Radiative Function Proof of the Lemmas

10 21

29 39

CHAPTER II Asymptotic Behavior of the Radiative Function

§s §6

§7

§B

Construction of a Stationary Modifier An Estimate for the Radiative Function Proof of the Main Theorem. Some Properties of r4
53

64 78 86

CHAPTER III Spectral Representation §9 §lO §ll §12

The Green Kernel The Eigenoperators Expansion Theorem The General Short-range Case

100 104 115 126

CONCLUDING REMARKS

135

REFERENCES

141

NOTATION INDEX

145

SUBJE CT INDEX

149

INTRODUCTION In this series of lectures we shall be concerned with the spectral representations for SchrBdinger operators (0. 1 )

T = -f';+Q(Y)

on the N-dimensional Euclidean space RN .

Here

Y = (Y ,Y , ... , Y ) N l 2 (0. 2)

N /} =

32

L -2

E

RN

(Laplacian)

j =1 8y. J

and

Q.(y) , 'w'/hich is usually called a "po tential", is a real-valued functim

on RN . Let us explain the notion of spectral representation by recalling the usual Fourier transforms.

The Fourier transforms N

f

(211)-2- l.i.m. R >m

(0.3)

where

fE L2(R~),

in the mean and Ys

F ± O

are defined by

e-ti YSf(y)dy

IYI< R

(E,1,t,2' ... , t: ) cR~, l.i.m. means the limit N is the inner product in RN , i.e .•

F, =

N (0.4)

yr.,

X y.E,. J J

j=l

Then it is well-known that

FOr are unitary operators from

L2(R~) and that the inverse operators FO± are given by

L2(~~)

onto

2

N

= (2~r)

-2

1. i .m.

RKO

(0.5)

for

F E L2(R~)

e +.iyr,F( ~)dr J IF,I< R t~

Here it shoul d be remarked that

'?

e±iy.;

are sol utions of

the equation N

(0.6)

'i'

f. j=l

that is,

e

±iy[

eigenvalue

>

r~) 'J

are the eigenfunctions (in a generalized sense) with the

1~!2 associated with -~.

Further, when

f(y)

is sufficiently

smooth and rapidly decreasing, we have FO-,(-M)(f,)

(0.7)

(-/\f)(y)

=

=

2 1r.:1 (F o::-f)(r,)

which means that the partial differential operator -6 the multiplication operator

1(:2 x

2

FO:t(1r,1 F ;1:f)(y) O

is transformed into

by the Fourier transforms

FO~.

Now we can find deeper and clearer relations between the operator -A and the Fourier transforms if we consider the self-adjoint realization of the Laplacian in

Let us define a symmetric operator

hO in

by

(0.8)

where

V(h O)

is the domain of

self-adjoint and its closure h . O

H is known to satisfy O

h O H = h O O

Then, as is well-known,

h is essentially O is a unique self-adjoint extension of

3

(0.9)

I

F*0 + x-· '.:>

t

-

../ 0

N H2 (R ) is the Sobolev space of the order ?, B is an interval in (O,co) , \/"8 is the characteristic function of IT = {(ER~/I(~2EB}

where

and

EO(o)

denotes the spectral measure associated with

tial operator sense.

-~ is acting on functions of

N H (R ) 2

H ' O

The differer

in the distribution

Thus it can be seen that the Fourier transforms

FO+

give unitary

HO and the multiplication operator !E.1 2x . Let us next consider the SchrBdinger operator T . Throughout this

equivalence between

lecture Q(y)

is assumed to be a real-valued, continuous function on

N R and satisfy (0.10)

Q(y)

Then a symmetric operator

-+

( Iy: • 'Xl)

0

h defined by (

D(h) (0.11)

{

hf

=

C~(RN )

= Tf

can be easily seen to be essentially self-adjoint with a unique extension We have (

(0.12)

l

i I

D(H) = D(H ) O

Hf

= Tf

(fED(H))

4

Here the differential operator T in (0.12) should be taken in the distributior sense as in (0.9).

These facts will be shown in f:12 (Theorem 12.1).

aim is to construct the "generalized" Fourier transforms

L2(R~)

operators from

f I (0.13)

where

j

FJHf) (r,)

l

E( B) = F* .. --- F t AlB -t

=

I" [2(F ,f)(()

F

~

H.

E::

2 L (R~)}

(f c V(H»

X/If are as in (0.9) and

associated with

F,- which are unitary

L2(R~) and satisfy

V (H) '" {f t- L2 (R~) 11;-:: 2 (F ,f)( t:)

Band

transforms

onto

Our final

•

is the spectral measure

E(o)

It will be also shown that the generalized Fourier

are constructed by the use of solutions of the equation

Tu = '~12 ~.:J u :

Now let us state the exact conditions on the potential lecture.

Let us assume that

Q(y)

(,y I ~

> O.

When

short-range potential and when potential.

E

>

0<

-r CXJ)

the potential E

<

The more rapidly diminishes

becomes the Schrtldinger operator T to easier.

in this

satisfies

(0.14) with a constant

Q(y)

Q(y) Q(y) -A

0.(y)

is called a

is called a long-range at infinity, the nearer

and

T can be treated the

In this lecture a spectral representation theorem will be shown

for the Schrtldinger operator T with a long-range potential

Q(y)

with

additional conditions

(0.15)

( Iy I

-r "" ,

j '" 0, 1.2, ... , m ) ,

O

5

j

D

where

is an arbitrary derivative of j-th order and mO is a constant determined by Q(y). Without these additional conditions the spectral

structure of the self-adjoint realization the one of the self-adjoint realization

H of T may be too different from H of O

-6.

The SchrBdinger operator

with a potential of the form

(0.16)

Q(y)

=

Th- sin

Iy I

for example, is beyond the scope of this lecture. The core of the method in this lecture is to transform the Schrtidinger operator

T into an ordinary differential operator with operator-valued co-

efficients.

1= (0,00)

Let

with its inner product space

L (I,X) 2

and

X = L (SN-l) , SN-l

( , ) x and norm

which is all

being the

2

X-valued

I Ix

(N-l)-sphere,

Let us introduce a Hilbert

functions

u(r)

on

I = (0,00)

satisfying DuRO2

(0. 17)

Its inner product

=

Joo0 lu(r)!x2 dr < 00 .

(, ) 0 is defi ned by

(0.18)

(u.v)O

=

(00 J (u(r) , v(r))x dr

o

N-l Then a multiplication operator L (R N)

(0.19)

2

3

f(y)

U=

-->-

r 2 x defined by N-l r 2 f(rw) E L (I.X) 2

(r=IYI ' can be easily seen to be a unitary operator from

w=

LIyl

E

S

N-l

L2 (R N) onto

)

L2 (I,X) .

6

Let

L be an ordinary differential operator defined by d

L :::

(0.20)

where, for each

T

2

-- + B(r) + C(r) 2 dr

(r

~

1)

> 0 , (x ex)

(0.21 )

and

AN

is a non-positive self-adjoint operator operator on SN-l . If polar coordinates

call~d

the Larlace-Beltrami

duced by

(j ::: 1,2 •...• N- 1)

(0.22)

1 !\N x can be written as

then

(0.23)

_ N~l ( . . )-2( .. )-N+j+l a L sln8 1 ···s 1 nG·_ 1 sln8. -.:::r;j:::l

J

J

ClUj

[(. )N-j-l ox ) slnG. ae- r l

J

j ,

C2 (SN-l) (see. e.g., Erd~lyi and al. [1], p. 235). The operator (0.23) with the domain C2 (SN-l) is essentially self-adjoint and its closure

for

XE:

7

is

The opera. tors

[IN"

operator

T a.nd

L are combi ned by the use of the un ita ry

U in the following way -1

(0.24)

H = U LU:>

for an arbitrary smooth function shall be concerned with tained for

Thus, for the time being, we

on

L instead of T,

and after that the results ob-

L will be translated into the results for

Our operator

T.

T corresponds to the two-particle problem on the quantum

mechanics and there are many works on the spectral representation theorem for

T with various degrees of the smallness assumption on the potential

o.(y).

L.et Q(y)

satisfy

(Iyl

(0.25)

+

C >

00,

0).

In 1960, Ikebe [11 developed an eigenfunction expansion theory for the case of spatial dimension eigenfunction ¢(y,[,)

(y,(,

N = 3

IR 3)

E

and

[>

2.

T in

He defined a generalized

as the solution of the Lippman-Schwinger

equation ( , , ) =e ¢lyE:

(0.26)

~(y,~),

and by the use of associated with

iyE, - 1

T.

4~

he constructed the generalized Fourier transforms

The results of Ikebe [1] were extended by Thoe [1] and

Kuroda [1] to the case where other hand,

Q(z )([) (z, r:) dz,

Iy-zl

N is arbitrary and

c

>

G(N + 1).

On the

Jager rl1-r4] investigated the properties of an ordinary differ-

ential operator with operator-valued coefficients and obtained a spectral representation theorem for it. N ~ 3 and the case of

E >

l.

His results can be applied to the case of

Saito [11,l21 extended the results of Jager to include

N ~ 3 and

[>

1,

i.e., the general short-range case.

At

almost the same time S. Agmon obtained an eigenfunction expansion theorem for

8

IR N with short-range coefficients.

general elliptic operators in

results can be seen in Agmon [1]. settled.

Thus, the short-range case has been

As for the long-range case, after the work of Dollard [1] which

deals with Coulomb potential Q(y) ;~' of

> ~

E

His

Saito [3]-[5] treated the case

along the line of Saito [1]-[2].

Schrodinger operator with as Saito [3]-[5].

>

E

Ikebe [2],[3] treated the

directly using essentially the same ideas

~

After these works, Saito [6],[7] showed a spectral

representation theorem for the Schrodinger operator with a general longrange potential

Q(y)

with

E

> O.

S. Agmon also obtains an eigenfunction

expansion theorem for general elliptic operators with lonq-ranqe coefficients. Let us outline the contents of this lecture. Throughout this lecture, the spatial dimension for each

r > O.

N ~ 3.

N is assume to be

As for the case of

N = 2,

Then we have

B(r)

~

0

see Saito [6], §5.

In Chapter I we shall show the limiting absorption principle for

L

which enables us to solve the equation (L - k2 )v

f,

=

(0.27)

(JL dr for not only

ik)v

+

0

k2 non-real, but also

(r

+

00),

k2 real.

In Chapter II, the asymptotic behavior of equation (0.27) with real

k,

range, there exists an element

(0.28) holds.

v(r)

~

v(r),

will be investigated. X E

v

e irk x v

the solution of the When

Q(y)

is short-

X such that the asymptotic relation (v

+

00)

But, in the case of the long-range potential, (0.28) should be

modified in the following

9

(0.29) with

Xv

E

X.

Here the function

A(y,k),

which will be constructed in

Chapter II, is called a stationary modifier. This asymptotic formula (0.29) will play an important role in spectral representation theory for

L which will be developed in Chapter III.

The contents of this lecture are essentially given in the papers of Saito [3J-[7], though they will be developed into a more unified and selfcontained form in this lecture.

But we have to assume in advance the

elements of functional analysis and theory of partial differential equations: to be more precise, the elemental properties of the Hilbert space and the Banach space, spectral decomposition theorem of self-adjoint operators, elemental knowledge of distribution, etc.

CHAPTER I LIMITING ABSORPTION PRINCIPLE Preliminaries.

§l.

Let us begin with introducing some notations which will be employed without further reference. IR:

rea 1 numbers

[:

complex numbers

[+ = {k Rek:

k + i k C [/k t 0, k > Of l Z l Z the real part of k.

Imk:

the imaginary part of

=

I

=

(0, «»)

T

=

r0,

= {r c:: R/

0

r

<,

k. < :Xl}

{r <= R/ 0 ~ r < ,"'}

ox;))

X = LZ(SN-l) with its norm LZ,s(J,X), s

E

R,

on an interval on

J.

I

'x

and inner product ( , )x

is the Hilbert space of all X-valued functions J

such that

(l+r)s!f(r) Ix

The inner product

f(r)

is square integrable

and norm

11 II

s, J

are

defined by (f,g)S,J

r (l+r)Zs(f(r), q(r)) dr

=

x

JJ

and

respectively.

I fll

s, J

When

s

may be omitted as in C~(J,X)

UC~(l~J)' where

= 0

or J

LZ(,J,X), J

II

=

lis

the subscript 0 or etc.

is an open interval in

I,

{y

r:=

RN/lyl c= J}

H6:~(J,X), s

E

R, is the Hilbert space obtained by the completion of

rl

J

=

and

N-l r--i-x

I

lJ =

is given by

(0,18).

11

C~(J

a pre-Hilbert space

X)

with its norn1

and inner product

B~2( r )

where ¢I

I

(r) :::

.?~p

When J ::: I

or

or s as in

FB(J,X), 3> 0, £

L

(B (r) )'2 with

:::

on

II

given by (0.21 ) and

B(r)

or

s

:::

0

we shall omit the subscript

B' H6,B O ,X) etc.

is the set of all anti-linear continuous functional

H6,B(J,X), i.e., J.

-I...

Hl,B(J 0 ' X) ~ :.;

V

l-

~? > E:c OJ,~ ~,v

,

such that III

iI ri { ii,

l} <

rs,J

00.

FS(J.X) is a Banach space with its norm III IIIS,J. When ~ = 0 J ::: I the subscript 0 or I will be omitted as in F(J,X), I!!

III J , I!! III:~ etc.

Cac(J,X),

J

being an (open or closed) interval, is the set of all

X-valued functions

f(r)

on

J

such that

f(r)

is strongly

absolutely continuous on every compact interval in exists the weak derivative f'(r)

E

L?(J ' ,X)

f'(r)

for almost all

for any compact interval

JI

C

J and there r

J.

E

J with

01

12

Cj(J,X)

is the set of all X-valued functions on

continuous derivatives. J

Here

J

having

j

strong

is a non-negative integer and

j

is an (open or closed) interval.

L2 (I,X)10c (H6,B(I,X)10c) is the set of dll X-vdlued functions f(r) such that ()f E L2(I,X) (pf ~ 116,B(I,X)) for any real-valued Cl function p(r) on T having a compact support in r. is the Banach space of all bounded linear operators from

m(y,Z)

into

Z with its operator norm

Banach spaces.

We set ID(Y,Y) ::: 113(Y)

denoted simply by V(W)

=

-AN +

I'Jhere

Y and

Z are

and the norm of ID(X)

is

II.

means the domain of

A = AN

II Ily,Z'

Y

W.

~(N-l)(N-3),

where

(as a self-adjoint operator in

AN

is the Laplace-Beltran; operator

X).

D = V (A) . 12

D = V(A\!).

C(a,S, ... ) denotes a positive constant depending only on

~,

3,

But very often symbols indicatinq obvious dependence will be omi tted. Hj(lR N) is the Sobolev space of the order L2 functions with order, inclusive.

j,

i.e., the set of all

L2 distribution derivatives up to the j-th

C~(lRN), L2U~), L2(D, (l+ly: )2s dy ) etc., will be employed as usual, where ~) Let

; s a measuab1e set ; n lR N.

L be the differential operator defined by

(0.20) and (0.21).

The remainder of this section will be devoted to showing some basic properties of a (weak) solution of the equation

2

(L - k )v

= f.

To this

13

end, we shall first list some properties of PROPOSITION 1.1.

[y

JJ

(i) Let

mN/!yl ~ J}.

E

H6,B(I, X).

be an open interval in

J

1 and let

Then

( 1. 1)

and ( 1. 2) where

(U y

U:~) B J ::: (y, '1J) 1

,

,

0

,. \J

( , )1 ,~.lcl

U is given by (0.18) and

is the inner product of

Hl U~J)' i. e. , (1.3)

(<,C

'~:)l,~J :::

( L,r {(I7.;)(y)

Co,TIm

•

)

+ y(Y)0r.YTj dy.

"'J

(ii) v

EO

Let

v

H6,B(I,X)

E

Ca/LX) n L2 (I,X)

(or

(or

v (" H6,f3(I,X)lOC)'

Then

v c Cac O,X)

L2 (I ,X)loc) with the weak L (I,X)loc)' Moreover v(r) cD';; 2 <-

derivative

Vi E

L (I,X) 2

(or

Vi E 1

for almost all

rEI

inner product

( , )B

with

I,

[32 v E:

and norm

L (I,X) 2 of

II 11[3

(or

B2 V c::: L (I ,X)loc)' The 2 H6,B(I,X) have the following

form:

rU'V)s

=

II {
+

( [3];; ( r)

(1.4)

,[3 l'2 (

r) v(r) )x

+ (u (r) , v (r ) )x}d r ,

[nulls (iii) (1. 5)

u(r)

Let

=

[(u,u)s]" v

c:::

H6,G(I,X)lOC' v(O)

Then we have the relations =

0 ,

14

!v(r) - v(s)i x

( 1 .6)

(r,s

Ii VII B, ( 0 , R)

E

10,Rl)

with (1. 7)

li VIl

[3,(O,R)

!v (r) 'x s /2

(1 .8)

£RQQF.

I v:1

[3 , (0, R+ 1)

(r E

(i) follows from two facts that

obtained by completion of

C~(~-:J)

n.

r 0, R

Hl(C;J)

is the Hilbert space

by the norm (1.3) and that the relation IX)

:IU..-:;I

[3,J

= 1:-;:1

holds for.;

1,:'J

obtained from the relation

Co. COU~J)'

(-~)~ =

-1

U

LOU~

The second fact can be easily , where we set

Let us show (ii) only for v E H~,[3(I,X), because the statement l about v E Ha , ,[3(I,X)l oc can be shown in quite a similar way. [3y the definition of H~,B(I,X) v c H~,[3(I,X) if and only if there exists a sequence

:¢n~

c

C~(I,X)

Cauchy sequences in set

lim

n HX,

,~I

-::

n

V

{On}' {¢~~, {[3~~n}

such that the sequences

L2 (I,X)

and

~n

converges to

v in

L2 (I,X).

are We

From the estimate

1 ,r

I'~n(r) - ~;m(r):X ~)o I¢~(t) - ¢~(t):Xdt

(1. 9)

it foll ows that the sequence r

E

fO,R]

'¢n(r)}

with an arbitrary positive

is an X-valued, continuous function on

is convergent in

R.

L

Therefore Letti ng (x

we arrive at

l.

v(r) n

-+ m

X uniformly for =

s-lim ¢ (r) nx:o n in the relation

X, 0 ; s < r

< ~)

15

(1.11)

whence follows that L2 (I,X).

Since

set e of

[3

v

CacCr,X)

c=:

and the weak derivative

C

I - e.

v

=

in

1

1•.

2(:,

n converges to v2 in L2 (I,X), there ex"ists a null such that B\1 ( r);) n(r) converges to v2 (r) in X for ,

r

Vi

v( r) c D-? with

Therefore

I

B':'(r)v(r)

because of the closedness of the operator

B~(r).

for

v2 (r)

-

r

- e

C

Thus we obtain

(1.12)

In a similar way we can show the first relation of (1.4) which is related the inner product

(u,v)B.

Finally, let us show (iii). on T such that JV G

as

H6,B(I,X), n",

(XC,

r(r) = I

~

(r

there exists

Take a real-valued ~

R + 1), = 0 (r

{on:

C

C~(I,X)

('" function

R + 2).

Then, since

such that

II '.'"v - '""'n II G r 0

whence follows that IV

(1.13)

-

"n I G,(O,R+l) ""'" 0

'"

Then, as is seen in the proof of (ii), r

E

[0, R + lJ.

n

=

1,2, ...

~n(r)

(n

~

"',) .

converges

(1.5) follows from the fact that

v(r)

¢n(O)

(1.6) and (1.8) are obtained by letting

n

= -+

i

=

II

I JS

r

I

<Xl

in the

L

¢~(t)dtlx ~ Ir-sl"21IQnIlB,(0,R) (0 < r, s < R)

and

in

X for

0 for all

relations (1.14)

p(r)

16

:,',.":' n (r) 2 X ~<' 2 ( I '",n r -,' nt, ~X t

(1.15)

.1.

(

)

,>,

(.,.)

,

'", 'I'

2) n (t) I.X

(0 < r ::: R)

respectively, where

I~;:

n

(t) I X;;

t err, r+ll ~~n (s)

mi n

r;s~r+l

has been taken to satisfy

! X·

Q.E.D.

The interior estimate for the operator PRQP~~TION 1.2.

function on

mN.

Let

and

t ~

Let

L is shown in the following:

L be as above and let Q(y)

v E H6,G(I,X)lOC

be a continuous

satisfies the equation

(1.16) with

k

exists

E [

C = C(k,R)

F(I,X).

Let

R be a positive number.

Then there

such that

(1.17) The constant set of

[x

C(k,R)

is boundp-d

~Ihen

the pair

(k,R)

I.

PROOF.

Since

v ~ H6,B(I,X)lOC'

we have by (ii) of Proposition 1.1

(d

Jd-r (v (r), (,)' (r) )X = (v' (r), ·f (r) )X + (v ( r), (1.18)

moves in a bounded

1l(v(r),

for almost all

l3(r){f,(r))x

reI.

1

l~ "

(

r ) )X

~

= (G~(r)v(r), B2(r)(~(r))x

Therefore from (1.16) it follows by partial

17

integration that

(1.19)

for any

$ E

co

•

fLlnctlOn on

i~(r) =

0

~

Set

=

~2¢n

Ilv - ¢n"B,(O,R+l) ~ 0

such that C

C~(I,X).

lO, R+2]

such that

(R+l S r 2 R+2).

( 1. 20)

( '0

R+ 1 2{

~

as

r:)n

n

is a real-valued

> 'Y,

o ;£

Then, letting

2 iv I IX +

2

;~: ~

and

;;I(r)

1,

~ ( r) ;::

n'~ 0'",

i

2' )

~ R+ 1 2

I

-2

'~. (v, ( k

f'::

we have

I [3"2V . X + I v I X~dr + 2 0 1

C<~((0,R+2),X)

in (1.19), where

R

1;,'.fi I (v I , v) Xdr

+ 1 - C) v) Xd r +
2 \!;

v> .

'0 (1.17) is obtained from (1.20) by the use of the Schwarz inequality.

Q.E.D. PROPOSITION 1.3 (regularity theorem). 1.2.

Let

with

k E [2

L be as in Proposition 2 be a "weak" solution of the equation (L-k )v

v E L2 (I,X)loc and

Let

f E L (I,X)loc' i.e., let 2

v

=

f

satisfy

(1.21) for all (1)

Q E C~(I,X). v

v

rEI

(2)

Cac(I,X)

E

for almost all

rEI.

with

v(r) c: D~2

[3v c:: L ((a,b),X) 2

1/

E

L?((a,b),X)

for each

for any

with the weak derivative B'2 V '

(1) ~ (4):

satisfies following

H6,[3(I,X)lOc ~'\ Cl(I,X).

E

for almost all Vi

Then

foY' any

v"

r? O.

v(r) E D

0 < a < b < and

Vi

0 < a < b

(r) <

00.

E

00.

D}2

18

( 3)

B'v

Cac(I,X)

C

with the weak derivative

(B-, (r) v( r) )

( 1. 22)

I

~ B\ ( r) v( r) + B'; ( r) v (r)

= -

I

holos for almost all rEI. (4)

We have 2 -v"(r) + B(r)v(r) + C(r)v(r) - k v(r)

( 1. 23)

for almost all PROOF. L2(ffi N)lOC

f(r)

rEI.

v = U-lv and f

Set

=

U-lf.

Then

-- f v,

belong to

and from (1.21) the relation

(1. 24 )

follows, where

( ')L

N L2(ffi ). As is (see, e.g., Ikebe-Kato

denotes the inner product of 2

N

well-known, (1.24) implies that

v c H2(ffi )loc llJ). Therefore there exists a sequence {~n} c c~(mN) N Y n ' v in H2(ffi )lOC as n '00 Set 1'n '" U';n· Then

H~,B(I,X)lOC in

as

L2((a,b),X) (0 obtain (1) ~ (4).

n ~ <

and

m

a

<

b

such that

:~n -, v in

{¢~}, {B6 n}, {B~¢~~ are all Cauchy sequences From these relations we can easily

< 00).

Q. E.D.

Now that Proposition 1.3 has been established, let us introduce one more function space.

Let

the set of all functions

v on

(b) :

be an open interval in

J

J

I.

DCl)

denotes

which satisfy the following (a) and

19

vEe 1 (J,X).

(a)

r

J

E

v(r)

L2 (J',X)

v(r) E D12 for each

v' , [32 v

D2.

f-

E

r

J. and

J'

in

For almost all B!2 v '

belong to

J.

with

Cac(J,X)

for almost all

E

Bv

for any compact interval

L

(b)

v'(r)

D and

E

r

1,

E

J.

It follows from Proposition 1.3 that the solution of the equation (1.21) belongs to

v

~

L2 (I,X)loc

H~,[3(I,X)lOC n D(I).

The unique continuation theorem for the operator

L takes the

fo 11 owi ng form. PROPOSITION 1.4.

Let

v

L be as in Propos it i on 1.2 and let

E

D(J)

satisfy the equation

2 (L-k )v = 0 on

open interval in

I.

Suppose that

v(r) = 0 in a neighborhood of some

Then

v(r) = 0 on

J.

r

point

O

E

PROOF.

J.

v = U-lv

k E lC,

where

as in the proof of Proposition 1. 3. is a solution of the equation (T-k 2)v 0 and v(y) = 0 for y

E

{y

E

Set

J with

]RN/1Iyl - rol

<

n} with some

rj

>

continuation theorem for elliptic operators to show that

v ~ 0 on

~Y ~ ]RN/ 1Y1

E

O.

J

is an

Then

-v

v

Thus we can apply the unique (see, e.g., Aronszajn ll])

J}, which completes the proof. Q.E.D.

Finally we shall show a proposition which is a version of the Relich theorem.

20

P-.RQ£.DS!J:JQJ'L__L."§.

Let

bounded open interval in Then

L be as in Proposition 1.2 and let J Let

1.

be a bounded sequence in

fv . nl

{v n} is a relatively compact sequence in

exists a subsequence sequence of PROOF.

IU } - m

of

such that

L2 (J,X), i.e., there is a Cauchy

L2(J,X).

vn a bounded sequence in

Set

u-lv n.

Then, by (i) of Proposition 1.1,

Hl(lJ)

with

~J

=

{y

E

mN/IYI

E

from the Rellich theorem that there exists a subsequence such that fU , m}

be a

is a Cauchy sequence in

is a subsequence of

Set

J}.

{v n} is

It follows

{ um)1

um = UU m.

and is a Cauchy sequence in

Then

L (J ,X). 2

Q.E.D.

21

§2.

Main Theorem.

We shall now state and prove the limiting absorption principle for the operator 2

d L'" - --+ G(r) + C(r) dr 2

(2.1)

which is given by (0.20) and (0.21).

The potential

Q(y)

is assumed to

satisfy the following t\~~VMPT I.Q.N

(Q)

l

(~)

can be decomposed as

Q(y) Q

~

.2: l. Q(y) =Q O(y) + Q (y)

are real-valued functions on JRN E

Cl(JRN)

~,

and

~Dj%(Y)1 ; cO(l + Iyl)-j-c

(2.2)

with

such that l N > 3.

with constants

0

~

C

JRN, j

=

j

0,1)

where

D

denotes an

sl > 1 and the same constant

Co

as in

Co >

0 and

(y

< E

1,

arbitrary derivative of j-th order.

(2.3)

with a constant We set

rC(r) (2.4)

=

CO(r) + C (r) , 1

1Cj ( r) " ~ ( rw) x

(j" 0, 1) .

(~).

and

22

(2.5)

f" Co(r)1

<

rCo(rl

l

<

cO(l+r(C , cO(l+r) -l-c.,

( r) II

< =

cO(l+r)

UC

l

where

-l-c l

denotes the operator norm of B(X)

II II

derivative of CO(r)

and

Cb(r)

in B(X).

Let us define a class of solutions for the equation k

E

[+

and

2 (L-k )v

=

t

with

£ c F(I,X).

QE£WITION 2.2. such that ~ given.

is the strong

<

(radiative function).

,5;; minC~(2+c), ~l)'

Then an X-valued function

.t?A."noticJn for

{L, k,

n,

Let

v(r)

Let f

on

E

I

be a fixed constant

0

F(I,X)

and

k

is called the

E

[+

be

!'cd·t:a1;ive

if the following three conditions hold:

H6' [3 (I ,X) 1oc'

1)

V F.

2)

Vi

3)

For all

ikv

-

E

L2 ,6_1(I,X).

(bE:C~(I,X)

-2

(v, (L-k ):b)O

we have

= .

For the notation used above see the list of notation given at the begin ni ng 0 f and hence

§ 1.

The condition 2) means

dV

jr

•k

1 V

is small at infinity,

2) can be regarded as a sort of radiation condition.

This is why

v is called the radiative function. THEOREM 2.3.

(limiting absorption principle).

Let Assumption 2.1 be

fulfill ed. (i) for

Let

{L, k, l}

(k, £) c [+ is unique.

x

F(I,X)

be given.

Then the radiative function

23 (i i)

For given

radiative function

there exists a unique v

=

v(o, k, £)

fl, k, Cl

for

wh1ch belongs to

Hal ,B ,(I ,X) . ,-0

(iii)

let

K be a compact set 1n

the radiative function for

!l, k,

there exists a positive constant

£}

let

[:+.

with

C = C(K)

k

v

v(o, k, £)

=

K and

be

F8 (I,X). Then depending only on K (and l) E

(E

such that "~: I II vB_'s II C ql,'-t'I" , 0 0

(2.6)

l

-

1,

llvll_

(2.7)

o + Ilv'-ikvll!)_l + IIB'~vIl8_l < CIII {' ili,~"

,

and (2.8)

(r

( i v) The ma ppin g:

[: +

x

r s (I , X)

~

1) .

( k, R) .. v( 0, k, {)

E

H1 , B, (I , X)

l;

[: +

is continuous on mapping from

a,-o

F8(I,X). Therefore it is also continuous as a F6 (I,X) into H6,B(I,X)loc'

[:+ x

x

Before we prove Theorem 2.3, which is our main theorem in this chapter, we prepare several lemmas, some of which will be shown in the succeeding two sections. lEMMA 2.4. {l, k,

Then

a}.

let

k~

and let

[:+

v be the radiative function for

v is identically zero.

The proof of this lemma will be given in :<3. For <{[f], (2.9)

f

¢> =

E

l2,e(I,X)

(f, ¢)a

for

with ¢

F

B>

a

C~(I, X) .

III f

[f]

III r,

we define {,[f]

C

Fs(I,X)

by

It can be seen eas il y that

2 I fll 3

24

LEMMA 2.5. {L, k, dflJ

k

~

E

L2 ,5(I,X)

Let

with

a compact set in

f

K and let

and

[+

0

v

be the radiative function for

and let

v

C

L2 ,_,)(I,X),

is as in Definition 2.2.

where

K is

Then

!.,

B2v

E

L2 ,8_1(I,X)

and there exists a constant

C

=

C(K)

such that

(2.10) and 2

(2.11)

II v I _5 , ( r ,

LEMMA 2.6.

Let

vn(n

(r>l) .

(Xl )

1,2, ... )

=

ik

I n

t

(2.12)

as

n -.

n

be the radiative function for

-+

k

~

.(

E:..

II v II

n

k , .en}, n

[+

in

and that there exists a constant

Xl,

~L,

F,)(I,X) such that

Co

,+ I Vi - iknvn"5_1 < Co , n

-l~

(2.13) 2

2 -(20-1)

II vn I -l':, , ( r,OO ) :; COr

for all

n

=

1,2, ...

Then

{vnJ-

has a strong limit

which is the radiative function for (2.14) as

n

v

n

-+ ()().

>

(r ; 1)

v in

fL, k,C. H1 ,G(I X) o ' loc

v

in

L2 ,_:\(I,X)

Further we have

25

LEMMA 2.7. with

S ~ O.

Then the equation

(2.15) has a unique solution

v in o

H6,B(I,X)

and

vo satisfies

Therefore

vo is the radiative function for fL, kO' £}. Lemmas 2.5, 2.6 and 2.7 will be proved in §4. LEMMA 2.8.

Let

V

E

o of the equation (2.15) with kE

[+.

H6,B(I ,X) n L2 ,o(I ,X) be a unique solution ,f c= F,(I,X) and kO EO [+, Imk O > O. Let ()

Then the following (a) and (b) are equivalent:

(a)

v is the radiative function for

(b)

v is represented as function for

PROOF.

v

= V

o + w,

;L, k, £}. where w is the radiative

{L, k, K!(k2_k~)vol}

Lemma 2.8 directly follows from the relation for

(2.17)

v E L2 (I,X)loc

-2 (v - vo' (L-k )¢)o

=

222 (v, (L-k )¢) - + ((k -kO)v O' Q)o

and Lemma 2.7.

Q.E.D.

26

By the use of these lemmas Theorem 2.3 can be proved in a standard way used in the proof of the limiting ahsorption principle for the various operators. PROOF of (I) (k, C)

E [+:<

K, v

F(I,X)

follol"is from Lemma 2.4. 'II "III 8 !',.(

Let us show the estimate (2.6), (2.7) and (2.8) with

replaced by

"f"

t (,

for the radiative function

L2 , _,,( .. I ,X) suffices to show C

The proof is divided into several steps.

The uniqueness of the radiative function for given

(II)

k

JtL~.9B.E;M__?3.

and

E::

f c L2 , (:,.( r ,X).

(2.18)

I V Ii -;)" = < CiI

because (2.6) and (2.8) with

l'ICII;s'.

from (2.18) and Lemma 2.5. each positive integer

v for

fL, k, fffll

In view of Lemma 2.5 it

fil.: ,

replaced by

Hil..:;

easily follol'J

Let us assume that (2.18) is false.

n we can find

with

k

n

v for

~

Then for

K and the radiative function such that

n

"v" n

(2.19)

,=1,

-r'

1

I f II" < --

n

,~,

= n

We may assume without loss of generality that

(n =

k n

1,2, ... ) -+

k

E

K as

n

-+

:Xl.

it can be seen that (2.12) and (2.13) in Lemma 2.6 hold good with t

= 0 and Co = 2C, where

we should note that

2.6,

i1f[f n ]lIC;

Thus

n = t[f n], C is the same constant as in Lemma 2.5 and ~ I

f nII (.~ -, 0 as

n ' "".

f

Therefore, by Lemma

fv , n} converges in

{L, k, OJ

L2 , -6 (I ,X) to the radiative function for which satisfies II vii -;5 = 1. But it follows from Lemma 2.4 that

v is identically zero, which contradicts the fact that we have shown (2.6), (2.7) (2.8) with

I vI

rep 1aced by

• = 1.

-0

II

fll

c)

Thus

for the

27

radiative function

v

E

L2 ,-0.(I,X)

for

{L, k.

~~rfn

with

k E K and

f c: L , 0(I , X) .

2

(III) kE

[;

The existence of the radiative function for

+ , Imk > 0

and

l

E

F,(I,X) l.

Let us next consider the case that and denote by with

Imk O >

(k, f)

has been shown by Lemma 2.7 with k

c-_

IR - .[ 0}.

Set

k

C = C(k O)'

(2.21)

n

=

vn the radiative function for {L, kn , i,'}. 0 and let Vo be the radiative function for

The existence of vn and Vo are assured by Lemma 2.7. (2.16) with B =~, we have

with

with

k + i/ n

G=

c.

E [; +

Let

kO E [;+ {L, kO' fl.

Then, using

whence easily follows that .2

IlvOI1_>.: (

", r,

110 '11 2 2 -(20-1)1,It: >.:

",) ~ C r

"

(r ? 1) .

wn = vn - V o is the radiative 2) vOl } . Since v and V belong to function for f L, kn , d (2 kn-k O o n 2 2 L2 ,LS(I .X), (kn-kO)vO and wn = vn - Vo belong to L2 ,c,(I ,X), and hence, as has been shown in ( 1T) , we have

On the other hand, by making use of Lemma 2.8, ?

(2.22)

C = C(K O) and KO = ~ kn f U{k}. It foll ows (2.20) , (2.21) and (2.22) that [k n } and fv } satisfy (2.12) and (2.13) with ;: n = R in n Lemma 2.6. Hence, by Lemma 2.6, there exists il strong limit v in

where

L2,_~(I ,X)

which is the radiative function for

{L, k, t'.

Thus the

28

existence for the radiative function for (IV)

=

va + w,

by lemma 2.8.

let

[+.

radiative function v

k and va be as in (III). O :L, k, f} (k c K, {E F,,(I,X))

v for

(~

let

K

Then the is represented

:l, k, {[(k2_k~)VO]}

w being the radiative function for It follows from (II) that the estimates

r

WII - + Ilw'-ikwll J

(2.23)

!I/wl/ 2 ~

hold with

has been established.

Finally let us show (2.6), (2.7) and (2.8) completely.

be a compact set in

as

{l, k, t}

<

-c,(r,<>:»:::

C::: C(K).

"8\'" 0 _1

o_1 +

2

2

; C'[ k -k 0 ',: II Vall (')

Cr-(2;~-1)lk2_k212 'va' II '1 2 0 0

,

(r :: 1)

(2.23) is combined with (2.20) and (2.21) to give

the estimates (2.7) and (2.24)

/I

v_II~, (r 00)

q,

.

,

~.;

r 2r - (2;; - 1) '" l'1 {'" I.I~: .2 v

(2.6) and (2.8) directly follow from (2.7) and (2.24). the mapping:

(k, f) ~ v(o, k,~)

in

H~:~6(I,X)

The continuity of

is easily obtained

from (I II) and lemma 2.6. Q.E. D.

29

§3.

Uniqueness of the Radiative Function.

This section is devoted to showing Lemma 2.4. the line of Jager [lJ and [2].

It will be proved along

But some modification is necessary, since

we are concerned with a long-range potential. We shall now give a proposition related to the asymptotic behavior of the solution v E D(J), J = (R,oo) with R > 0, of the equation (L - k 2 )v = 0 with k E R - {OJ. Here the definition of D(J) is given after the proof of Proposition 1.3 in §l. PROPOSITION 3.1. (J

(R,oo), (R;O) (1)

holds with

Let Assumption 2.1 be satisfied.

Let

v

E

D(J)

satisfy the followinq (1) and (2):

The estimate

k

E

R - {OJ

and some

c > 0,

where

6

is given as in Definitior

2.2. (2)

The support of v is unbounded, i.e., the set

~r

E

Jj

Iv(r) I x>O}

is an unbounded set. Then

lim{lv'(r)l~ + k2Iv(r)I~} > O.

(3.2)

r-;oo

The proof will be given after proving Lemmas 3.2 and 3.3.

Set for

vE:D(J) (3.3)

( Kv ) ( r ) = Iv'(r)12 + k2 !v(r)!2 - (8(r)v(r), v(r))

x

x

- (CO(r)v(r), v(r))

(rcJ).

x

30

Kv(r) is absolutely continuous on every compact interval in J. LEMMA 3.2.

Let

vED(J) and let (3.1) be satisfied.

c > 0 and l

R > R such that l

Then there exist

d -2~ dr (Kv) (r) ; - c (l+r) "(Kv)(r) l

(3.4)

(a.e. r ::: R ), l

where a.e. means "almost everywhere". PROOF.

(L-k 2 )v ~ f

Setting

and using

(3.1) and

(Q.l)

of Assumptior

2.1, we obtain 2Re{(v",v')

x

+ k2 (v,v')

x

- (Bv,v')

x

- (COv,v')x} - ddr ({[3(r) + Co(r)}x,x)x1x (3.5)

=

2Re(C l v-f, v')x +

~(BV'V)x

> -2c2(1+rr25Ivlxlv'lx

wi th

c2

2.1.

From the estimate

=

c + cO'

Co

v

- (CO v,v)x

+~(BV'V)x -

being the same constant as in

=

(Q.l)

(COv,v)x

of Assumption

and (3.5) it follows that we have with d -Kv dr

- (COv,v)x (3.7)

-28'(Kv) + (-r 2 - c (l+r) -28 ) (Bv,v)x l 2 + [~k2 c (1 +r r Iv I~ - ({ cl (1 +r )- 2oC 0 + CO} v, v )xl l =

-c

l

(l+r)

°

.:: -~(1+r)-2o(KV) + Jl(r) + J (r). 2

31

Jl(r) ; 0

for sufficiently large

for sufficiently large

o ( r ) II

C

too.

r.

= O(r -28') (r~)

By

r,

(~)

holds, and so

Therefore there exists

2r- l -

because

C

l

(1+r)-28

~ 0

of Assumption 2.1 Ilcl(l + rf 28Co(r) + J (r); 0 2

for sufficiently large

R > R such that (3.4) is valid for l

r ; Rl •

Q.E.D. Let us next set

I

d 1 2x} , Nv '" r{K ( e d v) + (rn 2--log r ) r- 2fl ev

( 3.8 ) where m

is a positive integer,

LEt1t1A 3.3. for fixed

Let

vED(J)

< fl <

t,

d

m ~ 1 l

and

w =. e dv.

dd (Nv) r

=.

= m(l_fl)-lr l - fl

»

(r '" R2 ' m = ml

(d/dr) (Nv)(r)

Then

is calculated as follows:

2 (Kw) + r Jr (Kw) + (l-21J)r- 2fl (m --foq r)

IWI~

- r-2fllwl~ + 2(m 2 - tog r)rl-2flRe(w~ w)x Iw '

(3.10)

2 1

x

- (Bw,w)

+ {k 2 + (1-2fl)r- 2]l(m 2 --fog r) - r- 2fl }

x

- (COw,w)

d

x

+ r - (Kw) dr

+ 2r l - 2fl (m 2 - £.og r)Re(w' ,w)x

Now we make use of the relations w' (3.11)

edv' + mr -l~"

-'I -fl-l -p d W w" = edv" + mr e v' + fIIr > w' - prnr

= edv" + 2mr- fl w'

Then

R > R such that 2 >

>

(Nv) (r) '" 0 Set

d(r)

=.

satisfy (1) and (2) of Proposition 3.1.

fl E (l/3, 1/2) there exist

(3.9) PROOF.

~

-

(]lmr-~-l + m2 r -2 fl ) W

Iwl2x

r

32 to obtain

Jr (Kw) = 2Re(w" + k 2w - COw - Bw,w')x - dd r

= (3.12)

({B + CO}x,x)xlx=w

d 2 2e Re(C v - f,w ' ) + 4mr- P lw' I 1 x

p - 2(pmr- - 1 + m2 r- 2).;)Re(w,w')

x

It follows from (3.10)-(3.12) that

d - ({CO + rCb} w,w)x + 2{re Re(C v - f , w' ) x 1 (3.13)

-2~mr-).; + (Bw,w)

+ r l - 2lJ -tog r)Re(w,w')

x

x

By the use of (3.1) and Assumption 2.1 we arrive at

dd (Nv) r (3.14)

~ r k2

2 21 + m (1_2l-1)r- l-

{(1-2p) -tog r + l}r- 2lJ

-

p 2 2 - 2c (1+r)-E j Iwl + (4mr 1- + 1) Iw'I O x x - 2{(c + c ) (1+rf(2o-l) + lJmr- lJ + r l - 2lJ ,£oq r} Iwl Iw ' I o . x x.

Thus we can find

R > R, 3

independent of

Jr(NV)

~ {~--k2

+

m,

such that

m2(1-2p)r-2P1Iwl~

l - 2 {2r - 2lJ ,£og r+ pmr-lJ}lwl (3.15)

x

Iw'l

x

2 1 1J + (4mr - + 1) Iw'I

x

=p (r,m) Iwl

2

2

' I + ' x - 2Q(r,m) Iwl x Iw x S(r,m) Iw x 1

33

This is a quadratic form of of

Iwl2x

Iw 1x and

l I I w IX

The coefficient

P(r,m)

satisfies

(3.16)

(r>O, m>O)

Further we have PS _ ~ > 4mr l - lJ p _ Q2 2mk 2r l -\-l + 4(1_2lJ)m3rl-3lJ _ {jl2 m2r - 2lJ + 4lJmrl-3lJlcg r + 4r 2- 4lJ (£og r)2} m2{4(1-2lJ)m _ jJ2 r -l+U}r l - 3]J + m(k 2 _ 4lJr- 2]Jlog r)rl-~

(3.17)

=

+ {mk 2 _ 4r l - 3lJ (log r)2}r l - lJ 2

(r;R 4 ,m;1)

> k /2

with sufficiently large

R4 > R3 ' R can be taken independent of m; 1 4

Thus we obtain (3.18)

dd (Nv) ( r) r

On the other hand that is, (3.19)

Nv

~v(r)

(a.e. r ; R4 , m ; 1)

; 0

is a polynomial of order

2 with respect to

m,

can be rewritten as 2 (Nv)( r) '" re 2d { 1 mr-lJ v+v 1X + k2 1v 1 X2 - (Bv, v) X - (COv, v) X I

I

+ r- 2]J(m 2 - lOR r) Ivl~} =

By

(2)

exists

- 2U I 1 2 re 2d {2r -2lJ1 v 12xm 2 + 2r -]1 Re(v,v' )xm+ ( Kv-r . v x£og r)}

in Proposition 3.1 the support of R2 ; R4 such that

v is unbounded. and hence there

3·1

Then, since the coefficient of there exists

m l

~

m2

in

(Nv)(r)

is positive when

r -= R ' 2

1 such that

(3.21 )

(m ;:;

In

l

)

(3.9) is obtained from (3.18) and (3.21). Q.E.D.

PROOF of PROPOSITION 3.1. a sequence

{tn}cJ, t l'CXl(n- KO ), n

Then there exists 3.2.

r

O Let us show that

(3.4) by

exp{c

l

J~o

~

First we consider the case that there exists such that

(Kv)(t ) >0 n

R such that (Kv)(r ) >0, l O (Kv) (r) > 0 for all r;:; r

(1+t)-2()dt},

R l O

for

n

=

1,2,

being as in Lemma In fact, multiplying

we have

(3.22)

whence directly follows

(3.23) It follows from (3.23) that

Iv· (r)

I~ + k2 Iv(r) I~

=

(Kv)(r) + (B(r)v(r), v(r))x + (CO(r)v(r), v(r))x

(3.24) > e

-clr~~+t)-20dt r

(Kv)(r ) O - Co(1+r)-Ck-2(lv'(r)l~ + k2Iv(r)I~) O

(r;; r ), O

35

which implies that (3.25)

Next consider the case that R5

>

R2, R2 being as in Lemma 3.3.

(Kv) (r) 2 0

for all

r:; R5 with some Then it follows from (3.19) and (3.9)

tha t (3.26)

which, together with the relation (3.27)

Jr Iv(r)

I~ = 2Re(v(r).

Vi

(r))x '

implies that (3.28)

with sUfficiently large support of

v(r) we have

R6 :; R5 .

Because of the unboundedness of the

Iv(R 7 )l x > 0 with some

R7 :; R6 ' and hence it

can be seen from (3.28) that (3.29) whence follows that (3.30)

Q.E.D.

36

In order to show Lemma 2.4 we need one more proposition which is a corollary of Proposition 1.3 (regularity theorem). PROPOSITION 3.4.

Let Q(y)

be a real-valued continuous function on

nN and let vE:.L 2(I,X) loc be a solution of the equaLion (3.31 ) kca;+ and

with

fC::L2(I,X~oc.

Then

222 lv' (r) - ik v(r) x2 ::: iv' (r) + k2v(r) Ix + kllv(r) Ix 2 2 + 4k l k211vII o, (O,r) + 2k l I m(f,v)O,(O,r)

(3.32)

1

holds for all

reI, where

kl '" Rek and

k2 '" Imk

PROOF. As has been shown in the proof of Proposition 1.3, v EO UH 2(R N) loc and there exists a sequence {.pn} C C~(RN) such that (3.33 )

as n

-+

00,

where

T;;;;;

-I",

+ Q.(y).

Set

¢:

n ::: U.p n

Then, proceeding as in the proof of Proposition 1.3 and using the relatior (L-k 2 )U

=

2 U(T-k),

we have dJ

(3.34)

r

v -+

v(r)

in L2(I,X)loc, in X (rEI),

-)0-

v' (r)

in X (r E !),

(L-k 2 )¢ -}- f n

in L2(I,X)loc

~n

-~

¢n(r) ¢~(r)

37

as n )-

frolll

(j)

0 to r

and make use of partial integration.

(¢~' ¢~)o,(O,r)

2 + ({B+C-k }¢n,¢n)o,(o,r) -

Then

I'Je

arrive at

(~~(r)'¢n(r))x

(3.35)

= (f n , By letting

n

-r

00

cPn)o,(O,r)

in (3.35) after taking the imaginary part of it (3.35)

yeilds (3.36)

(3.32) directlY follows from (3.36) and 2 2 2 2 lv' (r) - ikv(r) Ix = lv' (r) + k2v(r) Ix + kllv(r) Ix

(3.37)

Q. E.D.

PROOF of with

kE[+.

La~r1A

2.4.

Let

v be the radiative function for {L,k,O}

It suffices to show that

from (3.32), by setting

v = O.

Let us note that we obtain

f=O,

(3.38) Let us also note the relation (3.39)

1i m

:v'(r) - ikv(r)1

which is implied by the fact ttJdL

2

x

= 0

v' - ikVE=L 2 ,c_l(I,X)

then it follows from (3.38) and (3.39) that

38

(3.40)

~IIVIi2

which means

IIvl,O,(O, ex,)

that

k

2

2

= 0,

o,

0, ( 0, r )

r--><'"

i. e., k

=

=

0,

that is,

k{ lR - {O}

V =

o.

Next consider the case

Then from (3.38) and (3.39) we

obtain (3.41 )

~ r-}-CO

(

1

2 + k2 V (r) 2) '" .li!!.!..- v (r) - i kv(r) v (r) x ' x r·)-(X> x = 0 I

1

I

1

I

I

I

Therefore Proposition 3.1 can be applied to show that the support of v(r)

is bounded in

and hence

v = 0 by Proposition 1.4 (the unique

continuation theorem).

Q.E.D

39

Proof a r the 1cmmas

q. NO\'i

we shall shO\·J the lemmas in

!~2 ~Jhich

remain unproved. (k,~)

some more precise properties of the mapping

>

After that,

v::: v(·,k,,')

in

Theorem 2.3 will be shown. PROOF of LEt·l1-1A 2.6. with

v

= vn '

= kn

k

Applying Proposition 1.2 (the interior estimate) ~::: f

and

n,

we have

Ilvnl:B(O,R) ~ C(llvnlio,(O,R+l) + III ~nlll O,(O,R+l))

(4.1)

(ReI, {k n }

with

C = C(R),

{t }

are bounded sequences in

n

because

it can be easily seen that formly bounded for fore, for fixed

n

l:vn"O,(O,R+l)

and and

m -.

v in

n m

of

(4.3)

v

are uniR.

There-

By letting

m = 1,2, ...

in m -.

such that

vn

m

t~oreover, the norm

as

r -."

L2 ,_5(I,X) 'x,

in the relation

·--2 ) ( nm' L - knm):) '"

(v

is seen to be a solution of the equation

(4.4)

hl;:'nll : O,(O,R+l)

and

L2 (I,X)loc'

vn + v m ('t,).

respectively,

c

with a fixed positive number

(2.13), and hence we arrive at

as

F,(I,X),

{v n } and

the right-hand side of (4.1) is uniformly bounded

{v

to zero uniformly for

(4.2)

Since

Thus, Proposition 1.5 can be applied to show that there

exist a subsequence converges to

L ,_o(I,X) 2

1,2, ...

=

ReI,

n::: 1,2, ....

for

is a bounded sequence.

n=1,2, ... )

'I vn II -t' ,( r ,")

tends

m

by the. second relation of

40

Since (4.5)

u

=

v - v nm np satisfies the equation

(u,(L-k-2 )~,) nm

+

~"

n

-):

rn

(-~-

n

E

C~(I,X)),

p

Proposition 1.2 can be applied a9ain to show that

(4.6)

as

m,p

+

x,

whence follows that

(4.7)

as

m -)-

00.

Thus, letting nl

->- ",

in the estimate

(4.8) which is obtained from the first relation of (2.13), we see that

(4.9)

II

v' - i kv:1 6-1 , (0, R) -'~ CO'

and hence, because of the arbitrariness of Therefore

R > 0,

v is the radiative function for

v'-ikv c L2 ,o_1(I,X). {L,k,2}.

As is easily seen from the discussion above, any subsequence of {v n} contains a subsequence which converges in L2 ,_8(I,X) n H6,B(I,X)10C to the radiative function

v for

{v n} itself converges to

v in

[L,k,t1.

Therefore, it follows that

L2,_(~(I,X)

Let us turn to the proof of Lenlma 2.7.

n

(4.10)

Q.E.V.

To this end, we have to in-

vestigate some properties of the function space Hilbert space obtained by the completion of

H6,B(I,X)10C'

H6:~B(I,X).

C~(I,X)

It is a

by the norm

41

l '~~(I,X) c H1 ,8(I,X)1 HO oc , ~. 0 represented as

Obviously

and the inner product and norm are

(4.11)

Let

t

be a positive number.

~

Then the norm

"B,-B,t defined by

(4.12) is equivalent to the norm

(= II I!B,-I3,l)' The set of all antiB linear continuous functional on H6:~B(I,X) is FS(I,X), because we can

I I ,_(::3

easily show that the norm

\11£111

8

of

~EF,,(I,X)

1-

~"'j

is equivalent to the

norm (4.13)

I dJ. I B ,-G = l}.

It can be easily seen, too, that the norm

111Q,lli

3

is equivalent to the

norm defined by (4.14)

for all

t > O.

PROOF of LEMMA 2.7. on

Hl,B (I,X) 0,-8

(4.15)

\'ihere

Consider a llilinear continuous functional

Hl,B (I,X) O,-s

x

At

defined by

At[u,v] = (u,v)B,_g,t- 2Sf I(t+r)-2B-l(u,v )dr l

t

>

0 and

ko

E \1:+ with

positive definite for some

t

>

1m k > O. Let us show that At is O 0, i.e., there exists to > 0 and

42

IAt o1u ,u I .

(4.16)

j

2 C I' UI'1 B ;:: -'; t

' .',

It suffices to ShOrl (4.16) for (]I f 0)

and

c C'~(I ,X).

U "':

"'

C(r) = C(r) - 1 - ).

0

Set

2 k '"

o

J.+

1;;

Then we ha ve

(4.17)

By the use of a simple inequal ity

(a.!. 6)2

3 (1 -

,.t)i

+ (1 _

("x> 0,

a,o

(4.18)

~Atl(::',::)li2 ~}((:IOI:~,_~,t + Jr(t+r)-23(CQ'¢)xdr)2

l

IR)

0-

we have from (4.17)

+

jl2(Jr(t+r)-231~I~dr)2~

- 432{(Jr(t+r)-2R-1Re(O,¢')xdr)2 + (Jr(t+r)-2~-lrm(¢,¢I)xdr)2}

r (t) 1

can be estimated as follows:

(0<0<1).

Set

IX =

211(;11(2 1 (;11 + 1)2)-1

in (4.19).

Then we arrive at

1 )6 2

43

(4.20)

On the other hand, we have (4.21 ) (t-->-oo).

It follows from (4.20) and (4.21) that

~ E

Let f

E

Fs(I,X).

Hl,B,(I,X) 0,- 6

At

o

is positive definite with some

By the Riesz Theorem, there exists a unique

such that <~'¢> = (f,Q)B -~ t '

(4.22)

r 1

,

llif I! B , _ '{-"

t

0

~

C III

f-' 0

il~i

("

>,

where

C = C(B,t O) does not depend on f or Z. Now the Lax-Milgram Theorem (see, e.g., Yosida [11, p. 92) can be applied to show that there

exists

w E H6:~B(I,X)

such that

At [w , (jJ 1 = (f, ¢ ) B - (; t

o

(4.23)

'

j

II wll B _

, g,

t

0

f'S,

~ CII fll [3 _ '

0

t

8, 0

(ep

E

C~(I ,x))

(C--

By partial integration, we obtain, from (4.22) and (4.23), (4.24) ( 4.22 ) and ( 4.23 ) can be used again to see that

fies the estimate (2.16). tion for

{L,kO,t}.

Thus,

v

=

v

=

( to + r )

-2 ~

~w

(to + r )-2 Sw is the radiative func-

The uniqueness of the radiative function has been

already proved by Lemma 2.4.

Q.e.V.

.

satls-

44

In order to show Lemma 2.5, . . Ie have to prepdre hlo lemmas . LEt~i'1A

4.:.1.

Let

a positive constant

.

K be a cOnlpact set C

C(K)

=

11'+

1 n u,.

Then there exi sts

such that

(4.25) holds for any radiative function k2 > 0

and

f

Proof.

v

for

iL.k,tlfl}

with

k

Let

v,v'

Proposition 1.3 rc: 1.

all by

kCK

with

Imk:.:.k 2 >0

~nrl

v

(2-2~)Im

K,

and let

fC:L2.~(I,X).

v

=

v(·.k,qfj)

v

R to

E

((L-k 2 )v(r),v(r))x = (f(r),v(r))x T

(0 < R < T < cc)

and take the

Then

T

f

R

(l+r) 1-2o(v '.v)xdr - (1+T)2-25 Im (v' (T) ,v(T))

x

A T 2 2s 2 + (l+R) 2 - 2 "Im(v'(R),v(R))x - 2k k ~ (l+r) - ulvl xdr 2 l

(4.26)

= 1m

Since

Moreover,

integrate from

imaginary part.

E

1 B D(I) n H ' (I,X)loc by O 2 satisfies the equation (L - k )v = f for almost

L2 ,o(I,X).

E

r·1ultiply the both sides of

(1+r)2-23,

k + ik l 2

L2 ,o(I,X).

E

Then, by Lemma 2.7, there exists a unique radiative function such that

=

v,v'

E

fT

R

(l+r)

2-20 '(f,v) dr.

x

L ~(I,X), 2 ,.,

lim Im(v'(r),v(r))

=

lim(1+r)2-2c(v'(r).v(r)) = 0 holds. and r .~x: x 0 follows from (3.36). Therefore, letting T

+

ill

r->-O

along a suitable sequence k211V1121_

qn1

and

R ->- 0,

we have

~ ~ _1_ :(2-2(\) (I(l+r) 1-2l~lv' I Iv I dr + .(I (1 +r) 2-2 '~'! f Ix Iv Ixdr} (, 21 k I ' x x

(4.27)

1 -::_1_ {(2-2(~)J + J 1. 2 2 Ifk 1 I 1

Let us estimate

J

l

and

,J .

2

45

rJ l ,

(4.28)

{~llv'I,1

.,!lvll

-lj

l

;,:i- ~I;v'-ikvl; -0,+ ail vi! -c~}IlVlll

-d

(a = maxlk)),

5: -~;

kCK

) I

lJ 2 ~ II fill_til vi! 1-,';'

w~lere we used the Schwarz inequality.

Set

b:;; min !k l kCK

!.

Then it follows

from (4.27) and (4.28) that

(4.29)

~~ -dEwhere it should be noted that

2 C

function on

Let I

-~ <

:~-l

8-1

+ all vii

and

1-0

_<')

+ I fll ,S) •

<~.

(4.25) follows

Q.E.D.

directly from (4.29). LEr~t1A 4. 2 . ---

(II v '-i kvll

v

EO

D() "!'

such that

k,,11'+

and 0

\I,

~ ~~

1,

be a real-valued

•

and

(I' ~ R)

(4.30) (I'

with

R>

~

R+ 1)

o.

T T ~ !((l+r)o:-l jv'-ikvl2dr + (1 - ~21) !E,(1+r)ci- l I B\!2 dr R x - R x T T ~ Re !~(l+r)Q(f,v'-ikv)xdr

R

+ (4.31)

- Re !t(l+r)Q(clv,v'-ikv)xdr

R T

T

~ h;(l+r)'l(CoV,v)xdr + ~ hUl+r)0,-1(Cov,v)xdr T.

- k2 ~r.(Hrf(cov,v)xdr +

~ Jr.:' ( 1+1' )',./ i B\! ~

+ (CO v, v) ) dr

R

+

~ (l+T)~liv'(T)-ikV(T)!~ - (CO(T)V(T),v(T))x}'

46

f = (L-k 2 )v,

where

PROOF. (4.32)

Q

IR

E

The relation

and T ~ R+l. (L-k 2)v = f can be rewritten as

-(v'-ikv)' - ik(v'-ikv) + (B(r) + CO(r) + Cl(r))v = f

whence follows that (4.33)

-((Vl-ikv),v'-ikv)x - iklv'-ikvl; + ((B + Co + Cl)v,v'-ikv)x =

(f,v'-ikv)x'

Take the real part of (4.33) and note that 1 (frt d. v - "2 1

'

.

-1

k2

~

O.

Then we obtain

kv 2x:, + 2'1 crr 'I B\1 i,2x;- -"Z 1 ({dB (r )} v, v) x d iV ------ar 1

(4.34) + Re(Clv,v'-ikv) x % Re(f,v'-ikv) x . ~(l+r)~,

Multiplying both sides of (4.34) by

integrating from

R to T

and making use of partial integration, we arrive at (4.31). Q.E.D. PROOF of (2.10) of

LEr~r~A

L2,_c(I,X) be the radiative function for {L,k,SlfJ} with k E K and f E L2,6(I,X). Then v EO D(I) n H6,B(I,X)lOC by Proposition 1.3 and we have (L-k 2)v(r) = f(r) for almost all

rEI.

Let

2.5.

n =

Let

v

28-1 and

E

R = 1 in Lemma 4.2.

Then, it

follows from (4.31) that (6 -

"2)II~>(V'-ikV)!I~_l.(l,T) + (~- 8)"r;~2B~2v"Ll(l,T)

T

T

1

1

~ !t;:(1+r)28-1 :f:x!v'-ikvixdr + Co ff,(1+r)-2c!vfxlvl-ikvlxdr (4.35)

+ Co }r(1+r(26Ivi2dr + 26-1.. c 2

l'

+ Ck o 2

x

2

TJr'(1+r(2~lvI2dr x

0 1-,

T!f(1+r)1-2~lvl,2dr + 1 f2,;r !(1+r)2S-1;!B~vI2 + c ivl 21 dr ' x 2'" , ., x 0" x. 1

1

47

where

Co

is the constant given in Assumption 2.1 and the following estimates

ha ve been used:

(l+r)

28-1

(28-2-c ()-28 IICo(r)ll~col+r) ~cOl+r,

(4.36) (1+r)28-2 I1Co (r)1I ~ C (1+r)28-2-E-: ~ c (1+r)-28, o o I

l(1+r)26-1 IlCo (r)11

~

c (1+r)26-1-c o

~

c (1+r)1-20. o

By Lemma 4.1 and the Schwarz inequality, we have the right-hand side of (4.35)

(4.37) 1 26-1 ,2 + 2(1+C o)C1,3 !I vi, B, (1,2) +

~(1+T){(1+T)20-2Ivl(T)-ikV(T)I~

+

CO(l+T)-2tS!V(T)I~}

C; = maxlr:'(r)l. Since v'-ikv E L2,(~_1(I,X) and v E L2 ,_o(I,x). r the last term of (4.37) vanishes as T·~ w along a suitable sequence

with

{Tn:'

and hence we obtain from (4.35) and (4.37) I"

!lv'-ikvll o _l,(2"xJ) + iIB?vl!,~_1,(2,m) (4.38) -~ C{llvll_c') + II fil e') + Ilv'-ikvII 0 ,(1,2) + IlvII B ,(1,2)}'

(2.10) follows from (4.38) and Proposition 1.2 (the interior estimate).

Q.E.V.

48

PROOF of (2.11) of LEr:iHA 2.5_.

Frol'l (3.32) in Proposition 3.4, we have

(4.39) Multiply (4.39) by

(1 + t)-25

and integrate from

r

to

m

Then, it fol-

lows that 1 '~( )-2 'os I I • k ,2 d II vII, 2_ <5, (y ,x) :,'~ -k2 J 1+t V -1 v: X t 1 r

-----g-----

+

(Hr\J-(25-1)I'fll . IIVII A

! k1 ( 2"- 1)

(4.40)

~::

kj

.,

_8

...

(1+y-)-2(2:,-1)llv ' _ikvl ~-l,(r",,)

1

+ - - - 2---- - (1+ r )-(25-1)11 fl',1.,.11" v.I' _:". k !(2:)-l) -' , l

(2.11) is obtained from (4.40) and (2.10).

Q.f.V.

Now that all the lemnas in 52 and Theorem 2.3 have been proved completely, we can show more precise properties of the mapping (4.41 ) by reexamining the proof of the lenmas in §2.

...LEt1t:llL.i.).

Let

converges weakly to in

[;+

such that

Un} f

of

m

L ,rl:(I,X) 2

k .~ k l n

radiative function for {v - n i-.

Eo

be a sequence in

[; f-

as

with

{L,kn,trfnl!.

n k

¥

lX,

L , (\. (I,X) such that f n 2 Let ZknJ be a sequence

vn be the Then, there exists a subsequence l

[;+

as

n·' "".

Let

Jv , n} such that

(4.42) as

m ~ 00,

where

v denotes the radiative function for

{L,k,Z[f]}.

49

PROOF.

Since

is weakly convergent,

ff 1 - n'

'f

I

n1

is a bounded sequence

L2 ,6(I,X). Therefore, it follows from (2.7) in Theorem 2.3 that {vn~ is a bounded sequence in L2 (I,X)10c' that is, {vn~(vn = U-lv n ) is a bounded sequence in L2 (m N)10c' Noting that ~n satisfies the equation in

(4.43)

-tv n

'" U-lf )

{v --

is really a bounded sequence in

can see that

\'Je

n

n '

Then,

by the repeated use of the Rellich Theorem, it can be shown that there exists

{v n

a subsequence Therefore,

{v

E~ Hl , B(I X)

v -

0

'

mn loco

of which is a Cauchy sequence in m is a Cauchy seuqence in H~,B(I,X)lOC with the limit Moreover, it follows from (2.8) in Theorem 2.3 that

nm1 converges to v in L2 ,_t(I,X), too. In quite a similar way to the one used in the proof of Lemma 2.6, we can easily show that v is the

{V

radiative function for

{L,k,~[fJ:.

Q.E.D.

From the above lemma. we obtain Itl£QR~!.~.

(i)

Let Assumption 2.1 be satisfied.

Then the mapping

(4.44) is a

18 (L 2 • 8 (I ,X) ,L 2 ,_c(I ,X))-valued continuous function on [ if we set

+

,

that is,

(4.45) then in

(L - k2 )-1 c: 18 (L 2 ,0.(I,X),L 2 ,_()(I,X)) and (L - k2 )-1 is continuous k E [+ in the sense of the operator norm of 18 (L 2 ,6(I,X),L 2 ,_0(I,X)). (ii) For each k E [+, (L - k2 )-1, defined above, is a compact operatOl

50

PROOF.

Suppose that the assertion (i) is false at a point

Then, there exist a positive number {k n'-; c

II'

~

+

0

~

0 and sequences

kE

{f } C L

n

2,

[+.

C(I ,X)

such that

illf nIi 6- '-

(n = 1,2, ... ),

I

i

(4.46)

I ~

k -> k (n n II v( • , k,i. IfnO

> ~x~),

- v( • , kn ' ;~ Ifn /) I

With no loss of generality, we may assume that L2 ,:5(I,X)

to

f

~

e·

(n = 1,2, ... ).

f n converges weakly in

with

f c L2 ,(;(I,X). Then, Lema 4.3 can be applied to show that there exists a subsequence {n m: of positive integers such that fV(.,kn ,£[f n ]) m "

j

(4.47)

tv(·,k,qf in

L2 ,_o(I,X)

as

(4.48)

as

m7

m7

00.

1)

nm

-->

7

v(·,k,~[f]),

v(·,k,2rf])

Therefore, we obtain

IIv(·,k,i[f n ]) - v(·,k n ,£[f n 1)11 - j_ 7 0 m m m 00,

which contradicts the third relation of (4.46).

of (i) is complete.

(ii) follows directly from Lemma 4.3.

Thus, the proof

Q.E.V.

Finally, we shall prove a theorenl which shows continuous dependence of the radiative function on the operator THEOREM 4.5. ----(4.49)

Let Ln

=

Ln , d2

n

=

1,2, ... ,

- - - + B(r) + C (rlo

df

n

C( r) .

This will be useful in 56.

be the operators of the form C (r) = Co (r) + COl (r) n n

(r c= I)

51

assumed to be real-valued functions on

tively.

(4.50) with

('1

The constants

independent of

and

~

of Assumption 2.1 with

Q -1

n = 1,2, ....

which satisfy

replaced by

Co in

and

rn N (~)

and

~n

(t2.0)

and

respec-

and

(Q.l)

(Ql)

are assumed to be

Further assume that

1 im Q. (y) = QJ. (y)

n-+CO J n

l2.o(y)

and

Q.l(y)

which satisfy Assumption 2.1. {Ln,kn'~n}

be the radiative function for

such that

Let kn

vn ' +

E [,

n=1,2, ... , £n

E

Fo(I,X)

and k

l

(4.51 )

n

£n

as

n

-+

with

00

k

•

k

1n

-+ £,

in

-+

E [

+

and

11'+ \I..

.Q,

E

Fo( I ,X) • Then we have

(4.52) where

v is the radiative function for

stant

C such that

f

in

And there exists a con-

llvnIIB'-6'~CIIIQ'111 0'

I vn i knvnI u-" 1 + I B\ nI u-" 1 ~~ C \11 till 6' ) .;$ C2 r-(26-l) [II £111 (r $ 1) Lllv 112B _~ (

(4,53)

for all

{L,k,Z}.

n

=

I -

n

,\), r ,00

1,2,....

C = C(k)

8

is bounded when

k moves in a compact set

[+.

PROOF.

Let gn

be the radiative function for

{Ln,kO,t n} and 1et

+ wn be the radiative function for {Ln,kn£r(k~-k~)gnl}, where kO E [ , 1m kO > O. We have vn = 9n + wn by Lemma 2.8. Reexamining the proof of Lemmas 2.5 and 2.7, we can find a constant C, which is independent of

52

n=1,2, ... ,

such that L

(,I!g'!I, + i!B:!g :I . + n ~ n 0

! (4.54)

l:wn1

t II

for all

~n

II , 8

~

e :11£, nIII

i knvJ nI 6- 1 + II B\'w nII 0" 1~

-

"

t

e(II gn ll

+ I wnll _5:)

5:_

~)

u

II 2 ~ c2r - (2 ()- 1) (II I 2 + I w I 2 0) gn 8 n -Q wn -6,(r,oo) ,

n = 1,2,....

Thu s, we obta in, for all

ri! vn n i k vnII 6- 1+ I I

-

t

(4.55)

lin

vn11 2- c, . (r, co )

with a constant

C > O.

(4.56)

II wn11

I,

Ii B2 V nI \,5: 1 ~

,

(r ~ 1)

n

e( ill ~III

~e

2r - (25-1 ) ( II: 2111

~ U

5:

u

+ I wn~1 _,~), ~

~

+ II wnl! .0,) l;

The estimate

"~ell

-,j

9'n II

A v

(n = 1,2, ... )

can be shown in quitea similar way to the one used in the proof of Theorem 2.3 and Lemma 2.6. subsequence for

{L,k,O},

In fact, if we assume to the contrary, then there is a

{h m} of

{wn/I!w nII,} -(,

which converges to the radiative function

where we have used the interior estimate (Proposition 1.3),

(4.54) and (4.50).

l~e

have

II hll_ o = 1.

On the other hand,

h = 0 by the

uniqueness of the radiative function (Lemma 2.4), which is a contradiction. (4.53) follows from (4.54) and (4.56). 2.6, we can show the convergence of

Proceeding as in the proof of Lemma

which, together with (4.53), implies that

H~: ~ <5 ( I , X).

Q. Eo V.

L2 ,_o(I,X) ~ H~,B(I,X)loc' {v n} converges to v in

{v n } to v

in

Chapter II.

Asymptotic Behavior of the Radiative Function

§5. Construction of a Stationary Modifier Throughout this chapter. the potential

Q.(y)

is assumed to satisfy

the following conditions, which are stronger than in the previous chapter. ASSUMPTION 5.1.

(12)

Q(y)

Q(y) :::

~(y)

+ Ql (y)

121 are real-valued function on IR N,

and with (~)

can be decomposed as

such that

Qo

N being an integer

N > 3.

There exist constants Co m C O function and

0 < c < 1

> 0,

such that

~(y)

is a

(5.1)

where

DJ denotes arbitrary derivat1ves of jth order and

mO is

the least integer satisfying m > O

(5.2)

2_

mO ~ 3.

and

s

ml be the least integer such that mlc > 1. yJhen mO > 111 1 , we assume that (m, - 1)c < 1. Further ~(y) is assumed to satisfy Let

I

(Iyl

(5.3) (ill)

Ql (y) c

l

>

~ 1).

is a continuous function on max(2 - r:,3/2)

rn N

and there exists a constant

such that -E"

'Ql(y)1 5: CO(l + Iy:)'l

(5.4)

with the same constant REMARK 5.2.

(1)

Co

as in

(y

E

rn fJ )

(~).

Here and in the sequel, the constant

tion 2.1 is assumed to satisfy the additional conditions

0 in Defini-

~i4

whe n

l~

<: l' <

1,

when

0

(5.5) (l-l+i-l

--- -2 - ---'

<

l-

<

-2.

From the conditions (5.2) and (5.4), it is easy to see that (5.5) does not contradict the condition (2 )

If

( 3)

The condition

L > 1~,

~.

6 >

then

m ;;;; 3 and O

(ml-l)c

an integer, we can exchange

r.

<

1Il

;; 2 •

1

valued (4)

<X;

C

(l-(~(y) )%(y)

function on

is

f.

for a little smaller and irrational

The condition (5.3) is also trivial, because replaced by

In fact, when

1 is trivial.

such that

~(y)

and where

+ Q (y),

l

~(y) = 0

('.

21(y)

can be

'.J

is a real-

(!y! < 1),

=

l(

Iyl ,;'

2)

A general short-range potential -l-c Ql(y)

(5.6)

=

0)

O(IY:

does not satisfy (5.4) in Assumption 5.1.

In

~;12,

we shall discuss the

Schrodinger operator with a general short-range potential. The following is the nBin result of this chapter. THEORH15.3.

(asymptotic behavior of the radiative function).

Let

Assumption 5.1 be satisfied. Then, there exists a real-valued function 3 Z(y) = Z(y,k) on IR N x (IR - {O}) such that Z c=: C (lR N) as a function of y and there exists the limit (5.7)

F(k,J,) = s - lim e-h(ro,k)v(r)

in

X

r+<X>

for any radiative function 9, E

Fl+S(I ,X),

where

v for

p(y,k)

{L,k,Q.J

is defined by

with

k C IR - {Oi

and

55

)1

(y • k) = rk - \(y,k)

!

(5.8)

r

),(y.k)

with

r

=

Iyl,

w

J Z( t~;, k ) dt o

~

and

y/lyl

=

if

0<

ift
by making use of the next.

{L,k,,Q,}

with

k EO IR

and

- {O}

v be the radi-

Let

Assumption 5. 1 be satisfied.

Let

ati ve funct i on for

§7

t EO

Fl+S(I ,X),

S

>,-

being as in (5.9). Then we have u' - i ku, B2 U E L2 , S(I ,X), where u = e il- v and A is given by (5.8). Further, let K be a compact set in IR - {O}.

Then, there exists

C = C(K)

such that

(5.10) and (5.11)

for any radiative function where

v for

{L,k,~}

with

k

E

K and

1 E

Fl+S(I,X),

3 is as in (5.9). As was proved in Jager [31,

we may take

Z(y) _ 0 when

~(y)

_ O.

Sa i to [3] (and I kebe [2]) showed that we may take (5.12) in the case that approximation h of

Z(y) !;2

<

=

1 2k

E

~

Z(y)

~(y)

1.

It will be sho\'Jn that (5.12) is the "first

in the general case.

The function

A(y,k)

is

56

called a stationary

~E~if~e~.

In the remainder of this section, we shall construct the kernel of the stationary modifier lowing problem:

Z(y,k)

A(y,k).

To this end, let us consider the folFind a real-valued function >,(y,k) on IR N x (IR - {O})

such that (c>l.

(5.13)

for any xED, A(y)

~

where

~I

is given by (5,[1).

0 is a solution of this problem.

r-->-oo)

If C2.0(y):: O.

then

In order to solve this problem.

we have to investigate some properties of the Laplace-Beltrami operator '\N

on

SN-l.

Let us introduce polar coordinates

(r.3 ,9 2 •. ·.,3 N_l ) 1

as in (0.22), i.e.,

(5.14)

1,

(j

(5.15) = M.(e)

J

= 6.(e)-1 J

d dEL J

(j

(5.16)

A = -AN +

~(N-l)(N-3),

2,3 •... ,N-l).

1,2, ... ,N-l).

Then we obtain from (0.23)

and hence, setting

=

we have

57

(5.17) where

is a sufficiently smooth function on

(5.17) that the operator ~

on

D 2,

i. e., for

xE

M.

n

and

X

I

D~,

we define x

n

n

as

N-l

It fo 11 ows from

.

can be naturally extended to the operator

J

is a sequence such that

{x }

S

E

M.x by J

M.X J

s-lim M.X • where

x .~ x in

Cl(SN-l),

n-+<X)

J n

as

X

n

n -~al

M. will be considered J C2 function on IR N

In the sequel

+ m.

=

as the extended operator on and let us set

(5.18)

'P

= 'P(y)

=

.p'y'") \ '"

=

N-l r- 2 )" 01.\)2 j:=:l J

P

=

P(y)

=

P(y;\)

=

r -2 L".N \)

lM

=

N-l j1l

(r~

.!.) M..

J

J

r

LEMMA 5.5. Let xc: D and set with Z(y) E C2 (IR N). Then we have

$y)

(5.19)

= (
.1-

2,.-2 r M

fZ(tw)dt

o

(r

=

Iy!,

w = y/lyl)

+''p)(±i$ ex,

and (L- k2)( e iw x)

=

. -2 M) x + (iP + iZ I + C ) x eip{(B(r) + 2,r l

(5.20)

where with

- (2kZ - Co - Z2 ;p ,

P,

M are given in (5.18),

ZI

dZ

- ~-r'

and

-~

IJ (y) =

)xL rk - $y)

k E IR - {O}. PROOF.

(5.19) and (5.20) can be shown by easy calculation if we note

58

(5.21)

Q.E.D. In order to estimate the term P(y), we need LEt.f'1A5.6. Let h(y) EC 2 (IR N). Then (5.22)

N)~

(M.h)(y)

p;j

J

b (,.)-1 ,< Y j

P,J

U

"3h-

ay p '

(5.23)

3y . = --.;£. Let P,J d:;-!j as in (5. 18) . Then y

where

I p=l

. j (

'I

N

r

P(y) =

be as in Lemma 5.5 and let

fIb. (r; ) - 1

(M $ (y) = j

(5.24)

Z(y)

\ (y) ,

3h yp ely , . P

N

- (N - 1)

o p=j

J

rN -1 ~ rf 1.L

' )y __

- (N - 1)

be

~

_b Z I

dt ,

p,j GYpl ::.l y=tw

N

!

b.(A)-2 0 lJ=l p,q=l J

y

P(y)

[

2

yp,J.yq,J' ay6a;

l

P q_y=tw

dt' )~dt, ): ~y '8 1 p=l _p y~ y=ttuJ N

r = Iy!, w = y/lyl.

where

PROOF.

Let us start with N

(5.25)

()h

-- =

d8.

J

I"'

•

l..b....

L yp,J 3y , • P p=J

where it should be noted that from (5.25). (5.26)

yp , = 0 for ,J It follows from (5.25) that 82h

N

dej

p~j

-2 = -

:lh p c Yp

j y -=-{-- +

p < j.

(5.22) directly follows

a2h p,q=j p,J q,J aYpoYq

~

)

Y

.y

. -;---,,----.

59

Here we have used the relation oy

.

"

(5.27)

"1

a

2

y

~ '" '~2 0U. "In

.:j
(j::, p : N•

= - Y

p

at! j

J

Thus we obtain

N -2

L.b. p=J J

(5.28) N-l

N

()h

ypay

2

cos ':1. ah + ~ I bJ~ (N - j - 1) sin 8~ yp, j ()y , J p j= 1 p=j and, hence, we have only to show N-l

N

I

I

(

b"(S)-2~(N -

l

j :: 1 P'" j J

j

cos o. 1) --. ;;-1 y

-

s, n ,", j

(5.29) :: - (N - 1)

N )' y

}

. - yp adyh p ,J p

-~

p";l P 3yp

The order of summation in the left-hand side of (5.29) can be changed as N-l

(5.30)

N

I

l~ j =1 P'" j

::

N-l

8

J.

/,

p=l j::l

N

+

I

j=l

Therefore. it is sufficient to show J

::

P

8, b.(8)-2r(N_J"_1) /, J j:: 1 l

i

-----;co..--s----::s-><-j y _ y}:: sin oJ' p,j p

_ (N-l)Y p

(p '" 1,2, ... ,N-l)

(5.31) J

N

N-l = L\ b.((:I)j '" 1 J

To this end, let us note that

?{ (N-J'-l)

cos o. } J Y . - YN :: - (N - 1) YN . • s, n

C'J

c; j

p, J

P

GO (

I(

1

11------- -

o. cos_.J.. _

(5.32)

sin 8. y p,j

\sin2

=

(j
l;.

J

j

J

l-yp

(j = p ~,N-l),

whence follows that J

p

:;;

(5.33)

=-(N-l)y p The relation

(l~p-::N-l).

N = -(N - l)YN can be proved in a quite similar way, which completes the proof of (5.23)~ (5.24) follows from (5.22) and (5.23). Q.E.D. Let

$y)

I

be as in Lemma 5.5 and let DjZ(Y) 'O(IYI-j-,)

(5.34)

!

t2kZ(y) - %(y) - Z(y)

2

Z(y)

satisfy the estimate

(lyl +~,

j . 0,1,2),

-.r;(y) =O(iYI-E)

(1 < f: <

Iyl Then the estimates

(MjA)(y),(AN$(y)::: O(lyll-s)

from (5.24), where we should note that

(Iy! -)- m)

Ib j (8)-lYp,jl ~ Iyl

Therefore, it can be seen from (5.20) in Lemma 5.5, that

o(r-~)

holds good under the condition (5.34).

1

l+c

~ro).

are obtained

for j ~_ p. 2 i (L-k )(e Ux)l x

In order to obtain

which satisfies (5.34), let us consider three sequences {On(y,k)}, {An(y,k)} (yE lR N, wE sN-l, kE IR - {O}) defined by

-=

~(y), {~n(w,k)},

61

f:fl l (y)

1~1 (y) )'l(y)

-'0

1 2 k %(y),

= a

:0

¢ n+ 1(y)

1 r 2k- l%(tw)dt

(r;; Iyl,

= y/lyl),

('I

1 ;; 2 k f \20 (y) + (Q n (y) ) 2 +

.p (y ; An (y ) )}

(n (5.35)

o

1,2, .•. , m - 1) , O

=

(m

(n = 1,2, ... ,

2.3 •... ,m -l), l

-'0

ljI ((0) ;; n

I

r

IlAn(y) ;;

6 ~n(tw)dt

+ ~(r)ljIn(W)

(r ;; Iyl, ~(

where

) is a real-valued

n ;; 2, 3 , ... , m ) •

w;; y/lyl,

T

COO-function on

O

such that (0

I

O~~(r)~l,

(5.36)

,'(r) >

a.

and

,(r)

"11

(r .: 2),

and

mO,m,

for all

are as in Assumption 5.1.

n = 1,2, ... ,m -1. O

rnO ~ ml ,

If

then we set

ljIn(w) ;; 0

In the following lemma, it can be seen that

these sequences are well defined by (5.35). LEM~~

(y,k)

pair

tPn(y,k) ;; 0 ( i i)

5.7. E

(i)

9n (y,-k) ;; -(;n(y,k),

IR N x (IR - {a}) for

tP n(y)

E

and any

It' (w,-k) ;; -ljI (w,k)

n

n;; 1,2, ... ,mO'

Iyl ~ 1 and n;; 1,2, ... ,mO' m +l-n m +l-n C0 (IR N) and 'V n ( (J) E C 0 ( Sn- 1 )

n;; 1,2, ... ,mO• (iii) (5.37)

n

There exists a constant

C> 0

such that

for any

Further,

for

and (5.38) y c IR N,

hold for any

n::; 1,2, ... ,m

and j:: 0,1,2, ... ,m +l-n, where O O oj denotes an arbitrary derivative of the jth order. mO+l-n N (iv) As functions of k,

k moves in a compact set in

IR - {a}.

PROOF.

(i), (ii) and (5.37), (5,38) can be shown by induction. ~(y)

it should be noted that

:: 0 for

U~j h)( y) :: 0 ( Iy [1 -1 )

(5.39)

Iy! if

~

Here

1 and

Dh(y):: O(lyl-T)

(7 >O,lyj-->-oo),

If mO ~ m , then ~n(w) ~ 0 for all l and we may simply set \n(Y):: f~On(tw)dt. In the case

which follows from (5.22). n :: 1.2, ...• mO-l that

ml < mO' any logarithmic term does not appear in the estimation because (ml-l)s < 1. (iv) is clear from the definitions of Qn and

Q.E.D.

DEFINITION 5.8.

We set

Z(y) = Z(y,k):::I)

jA(y) :: A(y,k) :: A

(5.40)

mO-

lv(y) with r:: r y I, REt~ARK

Z(y)

E

mO-2

tAl::

=

(y) +

c,1(1")~

mO-

2(w),

r

2(y):: fZ(tw)dt, 0

V(y,k) :: 2kZ(Y) •. C20(y) - Z(y)2 - \O(y;'\)

y/ 1y 1•

5.9. (1) From Lemma 5.7, it can be easily seen that 3 N C (IR ), Z(y,-k) = -Z(y,k), Z(y) = a for Iyl < and

~n'

63

l'

j Iyl)-j-f.~ (D z) ( y) I ~ C(1 +

(y E IR N,

j = 0,1,2,3),

(5.41 )

lr (oj y) (y) I i

-j-{m -l)E O (yE: IR N, C(1 + Iyl)

j

= 0,1 ) .

Further, taking account of (5.5), we have

(5.42) where

S is given by (5.9). (2)

on

As a function of

IR = {a}.

The constant

in a compact set in ( 3)

k,

Z is a

3 C (IR N)-valued continuous function

C in (5.41) and (5.42) is bounded when

k moves

IR - {a}.

Let us consider the case that

~ < E ~

1.

Then ,

m0

=

3 and

(5.43) This case was treated in Saito l3] and Ikebe [21 (cf. (12) of Concluding Remarks) .

61\

~6

Let

Z(y)

. An esti ma te for the radiative function f, (y)

and

NO\v we are in a position to

be as in :.;5.

prove the first half of Theorem 5.4. Let us set for a func ti on cu(G)::: sup

(6.1)

G(y)

IR N

on

(1 + :y!)':'!G(y)

Y~IRN

The following proposition is an important step to the proof of (5.10) in Theorem 5.4. Assumption 5.1 be satisfied. Further assume has compact support in mN. Let K be a compact set in

PROPOSITION 6.1. that

r20(y)

IR - {O}.

~

Let

Let

be as in (5.9).

Then there exists

C::: C(K,Q)

such

that (6.2) holds for any radiative function

:L,k.~lfl}

v for

with

k c K and

L2 ,1+S(I,X), where u::: ei~v and ,(y) is given in (5.40). The j constant C = C(K.Q) is bounded when r ( n1) • r: .+ (D ~) , j::: 0, 1• 2 , ... , rn O' f

L

1.

are bounded. tive of

jth

1 -.

is given by (5.1). order and

J ['

v

DJ means an arbitra~y deriva-

are given in Assumption 5.1.

In order to show this proposition, we need several lemmas.

LEM~1A 6.2. 2 (L-k )v = f A(Y)

v

E

0(1) ~-'116,B(I,X)

k (' IR - {O}

and

f

E'

be a solution of the equation i L2 (I,X)loc' Set u = e \ with

defined by (5.40). (i)

(6.3)

with

Let

Then we have -(ul-iku)1 - ik(ul-iku) + Gu ::: ei).f - 2iZ(u l -iku) + (Y-Cl-iZI-iP)u - 2ir- 2Mu,

65

where

Y is as in (5.40) and

P,M are given by (5.18).

P

(ii)

Let

V(y)

and opera tors in

in

G. J

L2 (I , X) 1oc .

=

1. g.G. with Cl functions g. on

j=l J J X such that

J

Gju

E

Cac(I,X)

with

Then T

!(V{ul-ikuJ,ul-iku)xdt R

T

T

- !(Vlu,ul-iku)xdr + 2i !(ZVu,u'-iku) dr R R x T

(6.4)

+

'A

!(Vu,{Y-Cl-iZ'-iP}u + e' f) dr R x T

T

2

- J(Vu,Bu) dr + 2i J(Vu,r- Mu) dr} R x R x (0 < R < T) P (\g.

with

VI =

I

~ G.•

j=l ~r

J

PROOF.

(i) is obtained by an easy computation if we take note of Lemma 5.5 and the relation (L-k 2 )v = f. Take the complex conjugate of both sides of (6.3), multiply it by

Vu

and integrate on

SN-l

x

(R,T).

Then, by the use of partial integration, we arrive at (6.4). Q.E.D. LEt-1f~A 6.3. Let x, Xl E D and let S(y) be a Cl -function on IR N. Then (6.5)

PROOF

By partial integration, we have

(Sx,(M.j.)M.x l ) J

(6.6)

J

x

66

f

= -

S

N-l

: (t.1. . . ) ~1. (Sx)

,""J )-N-j-l + Sx b . (;",) -2 (s i n ...

J

.J

J

v ((sin ,u· ~ )N-j-l J

;) )-1 X }dw

• () f) •

-;;:;::,-

J

J

rJv"

whence we obtain (S ,~1x')x = -(t~(Sx) + Sx UlNA) ,xl)x (6.7) = -

(SMx,x')x - ((t~S)X,XI)X

This is what we wanted to show. LEMMA 6.4.

Let

~(y)

radiative function for where

B

V 1_ i

Then we have PROOF'.

Let

function for

Q.E.D.

be as in Proposition 6.1.

{L,k,2[f]}

isasin(5.9).i.e.,

2 r (spx.x ' )x·

=

with

3=~-f:

k E IR - {OJ

Let

v be the

and

f

(0
E

=0

L2 ,1+0(I,X),

(!~<E.:sl).

L

kv, [3"2 v c::- L ,,( I, X) 2 , IS i k = k + ri- (n = 1,2,3, ... )

n {L,k,1I.tl}.

vn be the radiative From Lemma 2.7, it fo 11 ows that and let

-((v~-ikvn)',v~-iknvn)x - iknlv~-iknVnl; + (Bvn,v~-iknvn)x (6.8)

+ (Cv nn ,v'-ik nnx v) by

~(r)(l + r)2G+l,

integrate from

0 to

r,( r)

T.

=

(f 'n vl-ikv nx )

being given by (5.36), take the real part and Then, proceeding as in the proof of Lemma 2.5

and using (5.5), we arrive at

67

1 2

( 8 + -)

2 1 T 2" J 2 T 2'~ !rl' f' ,J!G';; 0' ~"Iv'-ik n nv n I'dr x + (-2- - I'~) O,·r ! vn xdr 1

< =

J_( 1+T) 2 ~+ 1 Iv (T) - i k v (T) 2 n n n I

1

2 x

(6.9)

+

}t;(1+r)2r~+1Ifl Iv'-ik v o x n n n

1

dr, x

whence, by the use of Proposition 1.2 (the interior estimate) and Theorem 2.3 (the limiting absorption principle), it follows that L

(6.10)

IIv ' -ik v!1 + IIG' 2 v !I. < Cllfll +?, n n n B n 0~ l ~

with a constant

C which is independent of

n

~

(n

=

1,2 •... )

1,2, ....

Let

n

+

E:

L ,S(I,X). 2

00

in

the estimate (6.11)

Then it follows from Theorem 2.3 that 1

IIv ' -ikv!1 S.(O,R) + IIB' vI1 3,(0,R) : Cllflll+i~

(6.12)

2

I.

holds for any arbitrary

R > 0,

which implies that

v'-ikv,B 2 v

Q.E.D.

LEMMA 6.5. pact set in with

IR -

T > R+l > R > "

{Q},

Q(y) Let

be as in Proposition 6.1 and let v

be the radiative function for

fE:: L , 1+~ (I , X) , .-:'< being as in (5.9) . Let 2 where ;( r) is given by (5.36). Then we have for

k c K and

c,(r-R+l),

Let

K be a com{L,k,~rfl}

68

fT 2"" 2!. 2 2+ Cl~J::;.r "-:'(iu'-iku: + IB 2 uj )dr + Ilfli l +o + 'R x x p

(6.13)

l

-'I . 1-1 T. 2""- 1 n ~ -n- 1 + C2 !- k Re j'Jr"" (Z t1u.Bu)x dr • ,n=O R I 1m

I

~

i

where m l

u" e \ ,

r:(T)

-~

is a function of

is as in Assumption 5.1,

C R

T satisfying

lim n(T) '" 0, T->'" is a constant depending only on Rand

p = 1,2, j ;; 0,1,2, ...

,mO'

~1ultipl.Y both sides of (6.3) bY:tr 2;;.+1(U-''::-iku),

PROOF.

over the region K (6.14)

are bounded.

=

SN-l

x

(R,T)

and take the real part.

• ) I ") Re T ,"tr" ,u '-lku x r 2n+ 1 - (( U I -lku j-

R

+

(

integratf>

Then we have

Ru,u "-lku ) 'r d r

x

T 2('+1 " - Re ~(",r" {(eli.'f,u'-iku)x + ((Y - Cl)u,u'-iku)xJdr T

+ 1m hr

R

The left-hand side 1) K > ( ~) + '2 ro

2 ,~+ 1 f'

((Z' + P)u,u'-iku) dl" x

K of (6.14) is estimated from below as follows:

~k'YX 2J

2 T( nr 2 :'~ I B2. /2 d 1 - I),,) R u I -1"k U 1 xd r + ( 2' 'u x r I

I

(6.15)

K l

can be estimated as

(6.16)

Kl ;, C{llf ll l +"

RaY'21·;lu'-;kul~drFIIf{~+B+(R+l)1-25 Rl(;'lul~dr + lV~:o

69

where we integrated by parts and used (5.42) in estimating the T 2"+1 term Re J or ~ (Yu,u'-iku) dr. Here and in the sequel, the constant R x depend i ng on 1y on K and Q( y) . .Ii 11 be denoted the same symbol C. It follows from

(6.14) - (6.16) that

T 2«

2

2

1

c 1 l~r ~(Iu'-iku!x + \ B2u l x )dr

cn20+" 1 lu'-iku!x2 + T1- 2"c[u(T)l 2x +

~

(6.17)

In order to estimate Set

•. ) 6 . 3 ( 11.

V=

K 5, - }

2

2 and K3 in (6.17), we have to make use of Lemma 2B+1(Z' +P) in (6.4). Then "'r ~ K

Ren 22+1 ((Z'(T)+P(T))u(T) ,u' (T)-iku(T))x} + F(T) + G(R) 1

+ -k 1m

(6.18)

1

T

F(T)

and

G(R)

I' ('+ 1

!('cy.-,)

R T

+ -k 1m INI" R where

l

2')+1 R+l L 2 (1 + R) ,) 0' I B2u l dr x

2,) 1 1,-

(Z(Z'-+P)u,u'-iku) dr x ((Z'+P)u,Mu) dr, x

are the terms of the form I v II

2

~}

-0

•

(6.19)

C being a constant depending only on Rand K. Here, (5.17) is necesR sary to estimate the term ((Z' + P)u,Bu)x' Let us next set V = ~r2B-1M in (6.4).

Then it follows that

70

K3

~

1 2 1:_1 - k Re{T" U1u(T).u'(T)-iku(T))x: + F(T) + G(R) 2 T 2")-1 + k- 1m f.·,r' (Zt~u,u'-iku) dr I{ x 1 T 23- 1 , + k 1m Jar' (t~u,(Z +P)U)x dr

(6.20)

R T 2 i~-l

1

+ f Re [lr R Here we used the relation

(Mu,Bu) dr x (r 2/2) (P (u

Re(M(u'-iku),J'-iku)x

which follows from (6.5) with

x = x' = u'-iku

was also used in order to estimate the tl:rm

and

S(y)

Re(Mu,Yu)x'

=

1.

I -

i ku) •U

1 -

i ku) x

Lemma 6.3

Thus, we obtain

from (6.17), (6.18) and (6.20)

. . cr~ (6.21)

1m !ur2;j+l (Z(i"+P)u,u'-iku)xdr R

2 I m)~ :tr 2:'~-1(ZM U ,u "k) + -k-, u dr + R x

where we have used the fact that

Im~((Z'+P)u,Mu)x

+ (Mu,(Z'+P)u)x} = 0,

and

(6.22) 1-

L2 ,g(I,X) by Lemma 6.4 and the support of Z(y) is comIR N by the compactness of ~(y), it can be easily seen that

v'-iku,~2v E

Since pact in

1,

L2 ,(3(LX), which implies that lim 11(T) = O. J 2 and J 3 in T-~<x: (6.21) can be estimated in quite the same way as in the estimation of K2

u'-iku,B'2u

and

K 3

(6.23)

E::'

respectivelY.

By repeating thes2 estimations, we arrive at

T 21)

'ar JR

"'(Iu'-ikul I

2

x

2

+ IB' 2ul )dr 1

X

71

~ ~(T) +

+

F(T) + G(R)

rml - 1 n l T d L k- - Re J.-tr2i)-1(ZnMu,Bu)xdr

l

n=O

+ k

-m

+ 2k

R

1 1m

T 2~+1

far R

~

m (Z l (Z'+P)u,u'-iku) dr

x

T 2P-l m } 1m fCtr - (Z l Mu,u'-iku) d1' ,

-m l

R

x

m

is the term satisfying

Tl(T)

where

O(;YI-l),

lim rl(T) = O.

By noting that

Z(y) 1 =

"f-~

Q.E.D.

(6.13) is obtained from (6.23).

n Re(Z Mu,Bu)x

We have only to estimate the terms

in order to show Propo

sition 6.1 comrletely. LENMA 6.6.

Let

{

(6.24)

S(y)

be a real-valued

Cl

function such that

IS(Y)I~C

, : DS ( y)

1

.:

cr-

1

( y 1

1

~,

D =.iL.

where

1) (j = 1,2, ... ,N).

with a constant

c > 0,

(6.25)

1Re(S Mx,Ax) x ...<. Cr l - c ( iA12Xl2x +1 x: 2) x

aYj

'I

C = C(c,K,~) which is bounded when u

holds with

j = 0,1,2, ... ,mO' are bounded.

A ~ -AN

+ ~(N

PROOF. (I)

(r > 1,

c and

p

K is a compact set in

k

E

K,

j+c (0) "-Q ' IR - {O}

and

- l)(N - 3). We shall divide the proof into several steps.

By the use of (5.17)

J = Re(SMx,Ax)x

can be rewritten as

rN-l

J = Re(SMx,Ax)

(6.26)

Then the estimate

x

)

= Re{ )' (1-1 (SMx) ,~1 x) ~

In'';l

n

n

xJ

x

E

D)

72

Throughout thi s proof ...Ie sha 11 ca 11 a term

.

domlnated by ~ Cr

IMxlx

Cr

l-f:( IA!;i·2 I 12 ) xix + ,X'X

l-f:1 1~2 I - A x x'

for

Let us calculate

J

1

::

an

O. K, term when

J

K

is

C :: C(c,K,Q) .

Since

OK, term. 2 is an

Thus,

with

r .>- 1

we can easily show that

J .

we have only to consider the term

(II)

K

l

J • l

Re~N),l ((1-1 S)Mx,M x) } 'n='l n n x

~N - 1 N- 1 ) + Re~ (S )' (~1 r~.f.) (~1.x) ,r~ x) ~ In=l j';l nJ J n xJ

I

(6.27)

(N-l

N-l

In=l

J"'l

}

+ Re~ J: (S.l. (MJ.),)(Mnr~J.X),~1nX)x '" J 11 By noting that an O.K. term.

MnS(y)

J

Before calculating

13

(b~l J

M M. n J

(6.28)

{y E IRN/lyl ?._

is bounded on

-

M.M J n

::

,

,

P.

J

ll

is seen to be

we mention

cos '}.

-.··--7..JM Sln~J. n

(n > j),

J

(n = j) ,

0

\

I i -1 l -b n

('

cos .) n -.---M. J Sln C' n

which is obtained directly from the definition of

(n < j) ,

M. J

(see (5.15)).

Using

(6.28), we have

(6.29)

Here, Mx. n

J

132

we have

is an O.K. term, because, making use of (6.5) ...I ith

x :: x'

::

73

(6.30)

Therefore. let us consider J 12 + J 131 . (I I 1) Set

rZ (r~D)

(p = 1.2, ...• N).

=

! P

(6.31)

(j,n = 1,2 •... ,N-l).

where Y . p.J

obtain

__ ClY n

Then. setting cos ON

---t:.

dO.

and starting with (5.24), we

J

1 ~ - b- {b Z 1"1 j . j - j +1 j

(6.32)

=1

M

+

co s 6. . ~J cos G r p =3' +1 P P s, n v j p' , N

\

b Z

and

(j

r zIn

>

n).

N

(6.33)

t1

M.A nJ

=

1 Znn I

I \

Then

(6.34)

J

12

is rewritten as

b- 2 )' Z b cos 0

n p=n P p

cos d.J Z. + b~ 1 ___ (MnA) In J sin Q. J

P

(j = n).

(j

<

n).

74

122 is an O.K. term because Zjn(Y) = O(;yjl-~) by the first estimate of (5.34). Hence, in place of J 12 + J 13l l it is sufficient to consider Here,

J

(6.35)

with

(6.36)

r~ I

I

l (IV)

Now let us calculate

Using (6.32) and interchanging the

J'.

order of summation, we arrive at N

(6.37)

cos ""p n + Y p=n

b~2bpZpCOS ep '

and hence N

(6.38)

Z b cos p;;'l P P

which implies that

J' 1

op ,

is an O.K. term.

As for

J

2,

we can interchange

the order of summation to obtain J

2

=

~l-l N-l{ cos 0. Re !)' Y. (Sb: l --.--~-

Ln;l j=l

(6.39)

=

o.

J

Sln OJ

U1

A)(M.x),M x)

n

J

n x

75

Thus, we have shown that each term of J = Re(SMx,Ax) x is an O.K. term, which completes the proof. Q.E,D_ Proof of Proposition 6.1.

By Lemma 6.6, the last term of the right-

hand side of (6.13) is dominated by the term of the form F(T)

is given in (6.19).

sequence

{Tn}

Therefore, by letting

T+

00

F(T),

where

along a suitable

in (6.13), we obtain

(6.40)

where

C::: C(K,Q)

large in (6.40).

is independent of

R > O.

Take

R = RO sufficiently

Then it follows that

00

2 f r 2B ( IU'-ikuI + IB~uI2)dr ~ C{llfII 2 +lIv Il2 } + G(R) 1+B x x -D 0 R +1

(6.41)

o

(C

=

C(K,Q)).

C{lIf11 2 +lIv I12 .{'} by -u 1+B using Proposition 1.2, (6.2) easily follows from (6.41). Re-examining

Since

G(R O)

is dominated by the term of the form

the proof of Lemmas

6.2 - 6.6, we can easily see that

remains bounded when are bounded.

Pt:

(Q ) (DjO_) 1 'Pj+t: "'iJ' 1

C = C(K,Q) j ::: 0, 1 , 2 , •..

,m O

Q.E.D.

Now that Proposition 6.1 has been shown, we can prove (5.10) in Theo rem 5.4 . PROOF

of (5.10) in THEOREM 5.4.

Let

v be the radiative function

the compact set of lR - {O}, and Q, E Fl + , B + such that 1m kO > O. Then v where <5 - 1 ~B~l - <5. Let kO E a: can be decomposed as v ::: vo +w , where vo is the radiative func t i on for for

{L,k,Q,}

{L,kO'Q,}

and

with

W

k

E

K,

is the radiative function for

{L,k,Q,[f]},

76

It follows from Lenma 2.7 that

(6.42) iA

Set

Uo = e vO'

Noting that

and

(6.44)

with

C = C(K),

which follows from (5.19) in Lemma 5.5, we obtain from

(6.42) (6.45) Therefore, it suffices to show the estimate (~lO) with this end, we shall approximate

0o(y)

by a sequence

u

=

ei~w.

{Qan(y)},

To where we

set I

(6.46)

"-On

~(r)

and (r

n (y)

<

1).

= 0

r~~In (y) l n .! "{) I'

(n

= 1,2'00')'

is a real-valued, smooth function on =

0 (r ~ 2).

bounded uniformly for

T such that p(r)

Then it can be easily seen that n

=

1,2 •...

with

j = O.1,2 •...• m.

=

Pj+c(DJQon) Let US set

(6.47) and let uS denote by wn the radiative function for {Ln,k.£lfl~ with 2 f = (k - k~)VO (11 = 1,2, ... )_ For each Ln' the function z(n)(y)

is

77

~(y)

can be constructed according to Definition 5.8 with ~n(Y)'

replaced by

and we set

(6.48) Now, Proposition 6.1 can be applied to show I u I - i ku I

(6.49) with

n

C

=

formlyon as

n

-+

00

C(K).

Since

-+

00

+ II B 2 U I

IR N for each

(n = 1,2, ... ,).

< C {II fill + S+ I w I ..\'}

n S=

DjC!.on(y)

n

converges to

-u

Dj~(y) as n

-+

00

j =

un

-+

u in

H6~~(I,X)

as

n

-+

00.

Let

in the relation

(6.50) with

R > 0,

which is a direct consequence of(6.49).

Then we have

(6.51) Since

uni-

O,l •... ,m , it follows that )..(n)(y) -+ )..(y) O uniformly on every compact set in IR N. Therefore, by the use

of Theorem 4.5, we obtain n

1:

n S

R > 0 is arbitrary, we have obtained

(6.52) (5.10) follows from (6.42), (6.45), (6.52) and (2.7).

Q.E.D.

78

57.

Proof of the main theorem

This section is devoted to showing (5.11) in Theorem 5.4 and Theorem 5.3 by the use of (5.10) in Theorem 5.4 which has been proved in the preceding section. PROOF of (5.11) in THEOREM 5.4. v is the radiative function for

Let us first consider the case that

{L,k'~lfJ}

with kEK and f EL2 '1+S(I,X).

Using (i) of Lemma 6.2, we have for u=eiAv d 2irk 2ikr ~d e (u ' (r) -1 ku (r) , u( r)) x }=e g(r ), r 2 (7.1) , g(r)=lu'-ikuI'x2_IB\1 x-(e iA f,u) x+2i(Z(u'-iku),u) x -((Y-C,-iZ'-ip)u,u) +2ir- 2(Mu,u) . x x It follows from (5.10) that g(r) is integrable over I with the estimate :g(r) I dr~_I: u' -i kUll 2+il B\11 2+:1 fll ~ I vii x 0 0 " -u II 0

L

J

l

.t'

(7.2)

+C{1i u' - i kull Sll vii -6+ 1 vii :/11 B\:I } vii 2

.<:.C'IIf11 1+13

-6}

(s=8-do
Where C=C(K),C'=C(K) and we have made use of (2.7), too.

Integrate the

first relation of (7.1) from r to R (O
0

x

m'

x·

+ImI e 21k (t-r)9(t)dt. r By letting R-KO along a sequence {R n } such that (u'(Rn)-iku(Rn),u(Rn))x+O (n-7CQ ) (7.3) gives

'W} .

1 { II (u'(r),u(r)) 1+1 Ig(t)ldt iv(r)1 2 < Tj(T x ~ I~I m x )r Let us estimate Im(u'(r),u(r)). x Since

(7.4 )

d

d

dr {Im(u' (r) ,u(r))x ~"'Im{dr(u (r) ,u(r) )x} =1 ml(e - 2ikr-i r ,u ()) r x'lJ = Img () r , dr r,e 2irk( u '( r ) -1ok u () J

(7.5)

79

and (7.6)

I (U m

I (

r ) , u(r )) =(Z( r·) v( r) , v(r )) +I (V r ) , v(r )) ->-0 x x m x I (

(r-r0)

by (3.36), it follows that

IIm(u'(r),u(r))xl~ f:1g(t)ldt.

(7.7)

Thus we obtain from (7.4), (7.7) and (7.2) Iv(r)I~CllfI11+i3

(7.8)

with C=C(K).

Let

US

(rE I)

next consider the general case.

to be the radiative function for

{L,k,~}

v is now assumed

with kEK and

~EF1+P,(I,X).

Then, according to Lemma 2.8, we decompose v as v=vO+w, where Vo is the radiative function for {L,kO,Q.} with kd=(+' ImkO>O and w is the 2 2 radiative function for {L,k,£[ (k -kO)VOI}. It follows from Lemma 2.7 and (1.8) in Proposition 1.1 that (7.9)

1r Ivo(r)I~/2llvOII~CIII£1I11 l::

I VOl11 +B=
with C=C(k O)'

(rEI),

On the other hand we obtain from (7.8)

Iw(r)lx~Cllvo'I1+B

(7.10)

Q

+~:

with C=C(K,k )' O

(5.11)

(reI)

directly follows from (7.9) and (7.10). Q.E.D.

The proof of Theorem 5.3 will be divided into the following four steps.

Lemmas 7.1, 7.2, 7.4 and Corollary 7.3 will be concerned with

the asymptotic behavior of the radiative function v for {L,k,l!-j fl} witl kE: IH- {O} and LEM~1A

fE L2, 1+B(I

7.1.

, X) .

Let v be the radiative function for

kElR-{O} and fEL2 ,l+S(I,X). PROOF.

{L,k,~1

f]} with

Then Iv(r)l x tends to a limit as r-)-<XJ. Let RO>O be fixed. (7.3) are combined with (7.5) to give

rr

2 1

Iv(r)1 :;:~k.1 (u'(RO),u(R )) +1, g(t)dt O x m: R x m

(7.11)

) 0

+rX>e 2i k(t-r)g(t)dt}, )r

whence follows that (7.12)

J'"

limlv(r)12:::Mkll (U'(RO),u(R )) +1 g(t)dt!, O x mR x m O which completes the proof. r-+
Q.E.D.

LEMMA 7.2.

Let v be as in Lemma 7.1. o F(k,f):;:w-lim e-iM(r ,k)v(r)

(7.13)

Then there exists the weak limit

Y' ,","'

in X, where M(y,k):;:rk-A(y) and A(Y) is given by (5.40).

PROOF. Let US set

r (7.14)

.

-

1

~\(r)-2ike

~

- i M (r· , k ) ,

.

(v (r)+lkv(r))

:;:2~keirk(u'(r)+iku(r)-iZ(ro)v(r))

l ak(r):;:eirk(u'(r)-iku(r)) ==eirk+iA(ro )(v ' (r)-ikv(r)+iZ(r o )v(r)) with u:;:eiAv. (7.15)

Let

x~D. Then, by (i) of Lemma 6.2,

cJ( ex k( r ) , x )/: - 2he - irk (- (u ' - i ku ) , - i k(u ' - i ku )+i Z' u+i Z( u ' - i ku) , x )x :;:2~ke-irk{(u,Bx) -(ei/'f,x) +i(Z(u'-iku),x) 1

x

x

X

+( ( c1- i P- Y)u , x) x- 2i r - (u , Mx) x }:;:2 ~ ke- i rk 9 ( r , x) , 2

where we have used the relation (6.5) in Lemma 6.3 with x==u. x'=x and S(y)=1.

Similarly we have

(7.16)

~ r ( <\(r) ,x ):;: e irk [ ( u , Bx) x- (e iA f , x) +2i ( Z( U i ku ) ,x )x I -

-((Y-C -iZ ' +iP)u,x) -2ir- 2 (u,Mx) }==eirkh(r,x) 1 x x

1 E I.. Therefore, by (5.10) in Theorem 5.4 and the fact that IMxlx~Cr - IA2Xlx'

g(r,x) and h(r,x) are integrable over (1,00), which implies that there exist 1imits

81

(7.17)

' i

lim(ak(r),x) =a(k,x),

x

~::o

lim(&k(r),x) =&(K,x).

r->=

Here a(k,x)=O.

x

In fact, since u'-ikuEL2,~(I,X), there exists a sequence

{r n} such that lak(rn)l

tends to zero as n-7=. Therefore, by taking x account of the fact that (Z(r ')v(r) ,X) -)-0 as r-+<>o , which is obtained from the boundedness of Iv(r)1

x

x (rEI). it follows from the second relation

of (7.17) that -it!(r' k) lim e ' , (v'(r)-ikv(r),x)x r+ oo

(7.18) . {e -2irk (C£k - (r) ,x ) - 1e . -iJJ(r. 'k) ( Z( r' ) v (r) ,x )} =0 = 11m r-+ oo x x Thus we obtain from (7.18) and the first relation of (7.17) lim {(e r-+ oo

-iJJ(r. k) . -iJJ(r' k) ) . 'v'(r),x)x+1k(e 'v(r),x X}::21k ct(k,x),

(7.19) lim {(e-iJJ(r.,k)v' (r) 'X)x-ik(e-iJJ(r.,k)v(r),x)x}::O r-+ oo Whence follows the existence of the limits a ( k , X) = 1 i m (e - i r-+ oo

JJ (

r • , k ) v ( r ) ,x ) x

(7.20) -- - 1 l'1m (-iJJ(r.,k),() e 'v r , x)x i k

for xED. of I v(r) I

r-)-oo

Because of the denseness of D in X and the boundedness

x on I, which completes the proof.

Q.E.D. COROLLARY 7.3.

Let {RnJ be a sequence such that Rnt oo and

IV'(Rn)-ikv(Rn)I-+O as n-+ oo •

Then there exists the weak limit

(7.21) F(k,f) :: w-limak(R ) in X. n n-+ oo PROOF. Since {IV'(R )l .1 is a bounded sequence as well as n x {lv(Rn)l x }, it follows from (7.20) that the weak limit

82

,. -i/.t(R· k) w- n lID e n' v

I

Rn

(

)

ex i s t sin X and i seq ua 1 to i kF( k , f ) .

Therefore, (7.21) is easily obtained from the definition of

Q. E. D.

LEMMA 7.4.

Let v be as above.

There exists a sequence

n==1,2, ... , where CO>O is a constant, u=e iA v, and B is as above. 2.3.

Such a

surely exists by Theorem 5.4 and Theorem

{rn~

Let us set 1 ilJ(r o k) 2ik e (v'(r)-ikv(r)). t"

,

Then we have (7.25) v(r) == e i /-1(r ,k)();k(r) + e-iJJ.(r',k)u_k(r), o

the definition of ak(r) being given in (7.13), whence follows that 2 J:v(r )1 (7.26)[ n x

(ak(r ), e-iJJ.(rn·,k)v(r )) n

n

x

+ (Ct _ k( r n ), e +i JJ. ( r nO, k ) v ( r n ) ) x

an + bn

Here b n+0 as n+ oo because of the third and fourth relations of (7.23). Therefore, it suffices to show that a +IF(k,f)1 2 as n

n+ oo •

Setting

-iJJ.(r (7.27) Fn(k,f) = e n ' k) v(r n ), 0

we obtain from (7.15)

x

83

(7.28)

where (7.29) 9(r,x)

+ ((Cl-iP-Y)u-ei>-f,x) x x 4 2 + i(Z(u'-iku),x) + {-2ir- (u,Mx) } = I9.(r,x). x x j=1 J Now let US estimate 9(r,u(r n )). It follows from (5.5), (5.42) =

(Bu,x)

and the third relation of (7.23) that (7.30) 192(r.u(rn))! :; CO(~f(r)lx + C(1+r)-20[v(r)l x ) As for 9 3 (r,u(r n )) we have similarly (7.31) 193(r,u(rn)) I

< <

9 02(r).

Co C(1+r)-C-S((1+r)S(u -iku)l x CoC(1+r)-0~(1+r)3(ul-ikU)1 = 9 03 (r). x l

because

(7.32) -coB = {

(S=O),

-s < -6

-s-(o-c)

-0

=

(B=o-s).

94(r,u(r n )) can be estimated for r 194(r,u(rn))1

<

~

r n (.2':1) as

C(1+r)-2( 1+r)1-ElulxIA~2u(rn)!x

C(1+r)-1-c(1+r )~-Slul 1(1+r )r:+~B!2(r )u(r )1 n x n n n) ~ CCo(1+r)-E-S-~lvlx ~ CI(1+r)-~-a = 904(r),

(7.33)

=

wher e a = 0 (0 < s ';:]"2),

=

c (12< £ ~ 1 ) •

Her ewe ha ve use d (5. 11) i n

Theorem 5.4, (7.32) and the first relation of (7.23). enter into the estimation of inequality as ~ ~(a2+B2) -

9(r.u(r n )).

Let us

We have from the

2

2 ~ 2 191(r,u(rn))1 ~ ~IB2ulx+~r- IA 2u(r n )l x < 12 1(1+r)BB12uI2+~r-2IA~u(r )1 2 (7.34) x n x - 901(r)+~r-2IA~u(rn) I~ ~

4

Set 90(r)

=

I 9 ·(r). j=1 0 J

Then 9 0 is inte9rable and it follows

84

dr

2r

Let as

R~oo

in (7.35) along a sequence such that Iv'(R )-ikv(R ) I n

n

x

~o

Then, using Corollary 7.3. we arrive at

n~oo.

nn-f

XJ

(7.36) lan-(F(k.f),Fn(k.f))x l -'S

rn

go(r)dr+Jq r n IB!2(r n )u(r n )!

By the second relation of (7.23) and Lemma 7.2 the right-hand side of (7.36) tends to zero and (F(k,f),Fn(k.f))x coverages to 2 IF(k,f)1 as n~oo. which completes the proof. x

Q.E.D.

PROOF of THEOREM 5.3.

Let v be the radiative function for

{L,k,.Q,; with kE IR-{O} and .Q,c:F,,(I,X).

According to Lemma 2.8

~-<)

v is rewritten as v=vo+w. where V is the radiative function o for {L.kO'.Q,} with kO€C+. ImkO>O and w is the radiative function for {L.k,.Q,[f]} with f=(k2_k~)vO.

Since VOEH6,B(I.X). it can be

easily seen that (7.37) s-~lm vO(r)

0

In fact. letting

in the relation

R~w

2

2

fR

,

(7.38) Ivo(r)l x '" IVO(R)l x - 2 rRe(vo(t)'Vo(t))dt along a sequence {R } which satisfies IvO(R )lx~O (n-)-C<». we have n

(7.39)

1

n

21m I 2 vo(r)l x ~2 rlvO(t)lxlvo(t)lxdr ~ IIvOIIB.(r.m),

whence (7.37) follows.

As for w we can apply Lemmas 7.1. 7.2

and 7.4 to show (7.40)

W-lim e-iJ.1(r .• k)w(r) = F(k,f) in X, {

r~(X)

~im Iw( r ) Ix

=

I F( k • f) Ix '

85

which implies the strong convergence of e-iM(r.,k)w(r) Therefore the existence of the strong limit F(k.£)

(7.41 )

= s-lim e-iM(r.,k)(vo(r)+w(r)) r-+ oo =

F(k,f)

(f

=

2 2 (k -kO)v O)

has been proved completely. Q. E. D.

86 _ - i 'j 11m e I· v (r)

Some properties of

38.

r-K>'J

In this section we shall investigate some properties of s - lim e-ip(r,k)v(r), function

v for

F(k,£):

whose existence has been proved for the radiative

{L, k,

t}

with

kE: R -

{O},

~c

Fl+S(I,X).

The

results obtained in this section will be useful when we develop a spectral representation theory in Chapter III. LEMMA 8.1.

Let Assumption 5.1 be satisfied and let

F(k,£)

be as in

Theorem 5.3, i.e., F(k,£) = s - lim e-i~(r.,k)v(r) r-+«

(8. 1 )

where

v is the radiative function for

where

B is given by (5.9).

{L, k,:d

in X,

with

k E F + (I,x), l 3 Then there exists a constant C = C(k) such

that (8.2)

C(k)

is bounded when PROOF.

k moves in a compact set in

R -

[O}

As in the proof of Theorem 5.3 given at the end of §7,

is decomposed as

v

= vO+w.

v

Then, as can be easily seen from the proof

of Theorem 5.3, (8.3)

IF(k,£)1

x

limlw(r) I

\'"'"""K"J

X

90 : (k 2 - k2 ) v ' k being as in the proof of Theorem 5.3. O O O follows from (3.32) with v:w, f=gO' kl=k, k2=0 that

Set

It

87

Iw(r)l~ ~~;{lwl(r)

- ikw(r)l: - 2k 1m (go,w)o,(o,r)}

~~

~r ClIgolI~

(8.4)

;

with

C; C(k)

have been used.

~

and Let

Iw ' (r) - ikw(r) I: + 1Wi

(r) - i kw (r) 1~ +

C' = C'(k).

lh

IV'(r n ) - ikv(r)1 -)- 0 as n x

n-+=

Q, III ~

Here (2.7) in Theorem 2.3 and Lemma 2.8

along a sequence

r~~

C1 III

.

{r n} such that

r too and n

Then we obtain (8.2). Q.E.D.

Since the radiative function respect to

v(o,k,Q,)

fOr

{L,k,£}

is linear with

which follows from the uniqueness of the radiative

Q"

function, a linear operator

F(k)

from

Fl+G(I,X)

into

In the following lemma the denseness of

Fl+p,(I,X)

in

X is well-defined

by F(k)t ;: F(k,Q,)

(8.5)

shown, which, together with Lemma 8.1, enables to a bounded linear operator from LEt~MA

Q,

l

8.2.

Le to:::

.p ::: ~~ •

US

into

Fo(I,X)

to extend

will be

F(k)

X.

r B(I , X) is dense in F

Then

PROOF.

As has been shown before the proof of Lemma 2.7 in §4,

F (I,X)

can be regarded as a bounded anti-linear functional on

H6:~~(I,X)

which is defined as the completion of C~(I,X)

By theORiesz theorem there exists w;

uniquely

Wn E N

Hl,B (I,X) O,-.p

by the norm

such that

88

(8.7)

< .Pv, v)

where

( , )B,-lr?

R

< Q. ,

~(t)=l

wn(r):: tiJ(r-n)w(r).

IIw-wnIlB,_
-+

0

as

~(t)

Let

[3, -<;

(t~O),

be a real-valued, smooth (t~l),

=0

The norm

and set

v)

n

with

Ilwll

such that

(8.8)

Hl,B (I,X) O,-.p

is the inner product of

is equi val ent to function on

:: (w , v ) B, -
n-)
(8.9)

and we have

which implies that

III.Q, -

Thus the denseness of

.Q,n c= Fe(I,X)

Obviously

.Q,

n III .p

F~(I,X)

in

-+

(n-+=)

0

F (I,X) .;

(O;.p~B)

has been proved.

Q.E.D. From Lemmas 8.1 and 8.2 we see that the operator

r(k). k

can be extended uniquely to a bounded linear operator from X.

F (I,X) 8

into

Thus we give DEFINITION 8.3.

extension of

We denote again by

F(k) . When

(8.10)

.Q,::.Q,/fJ with

Hk).Q,1 fl

Now we shall show that radiative function .Q,

R - {OJ ,

EO

C

F()(I,X)

H6:~o(I,X) +

f ~ L ,o(I,X), we shall simply write 2

= F(k)f

F(k).Q,

v = v(o,k,.Q,)

F(k) the above bounded linear

for

[L,k,Q,} .

and the

As we have seen above,

can be regarded as a bounded anti-linear functional on On the other hand any radiative function

(kEa:,Q,E F (I,X)) 8

belongs to

H6: ~\) ( I, X) .

Therefore

well-defined, and we have

(8. 11 )

£

can be represented by

(.Q"v)=lim<1"v)

n· ~<X)

n

,

v

for

<.Q" v )

{L, k,.Q,} is

89

where

v

n

H6,B(I,X)

E

THEOREM 8.4.

v

in

H6:~o(I,X).

Let Assumption 5.1 be satisfied. {L,k,~.}

the radiative function for (j=l,2)

vn'~

satisfies

with

J

Let

vj (j=l,2)

be

~. E F"(I,X)

k E R - {O},

J

0

Then

(8.12) where the right-hand side is well-defined as stated above and the bar means the complex conjugate.

IF(k)~!~=-~

(8.13) fOr the radiative function ~

E

In particular

F (I,X).

When

6

v

for

Q'j = ~,ffjl

Im('J."v)

{L,k,~}

with

with

k

E

R - {O} and

f j to: L ,c(I,X),j=l,2, 2

(8.12)

takes the form (8.14) Further we have (8.15)

for the radiative function

PROOF. where

(~l' ¢) (8.16)

Let

US

E

for

[L,k,Q, If]}

with

C~(I,X)),

E

R - {O}

and

~jC Fl+S(I,X), j=l,2,

Starting with the relation

2 (v l ' (L-k )CP)O

we obtain

Ir {(v 1,4l )x + (B;'?vl,Bl:~~»)x + ((C-k 2 )v l ,tP)x}dr I

k

first consider a special case that

S is given by (5.9). (¢

v

=

(~l'ep)

.

=

90

Let

p(r)

O~p(rhl,

be a real-valued

p(r)=l (r~l),

(8.17)

Pn(r)

Cl

=0(r;2)

function on

such that

and let us set

p(.~.)

=

I

(n=l ,2, ... )

Then it can be easily seen that r < n or

if

(8.18) (r

E

I,

r> 2n ,

2"1 c =ma x I p (r) I ) • I

rE:I

Substitute

(8.19)

¢ = Pnv2

II [r~(vl

in

(8.16).

Then we have

2 'V 2 )x + Pn{(V,'V 2)x + (B\l,B\2)X + ((C-k )v =

Quite similarly, by starting with

l

V2 )x}]dr

< ,Q,l' Pn v2)

((L-k 2 )¢, v ) = < x'2'¢)' 2

it follows

that

(8.19) and (8.20) are combined to give 2n

(8.21)

In f)~{(Vl'V2)x

- (v 1,v 2)x}dr = < Q,2,P nVl ) - < Q,l,r: nV2)'

Since (v l (r),v 2(r))x - (v1(r), v (r))x 2 (8.22)

•

= (vl(r), v 2(r) - ikv (r))x - (v1(r) - ikvl(r), v (r))x 2 2 - 2ik(v (r),v (r))x l 2

91

and (8.23) with

h(r)-+O

as

rroo

by Theorem 5.3,

2n (8.24)

In

p~(r)dr ;

we obtain, noting that (8.18) and

j2n

I

IPn(r) ,n

=-1

(8.25)

< C

2n{ lblrdl In

+

(r

-0

Iv1 Ix )( r"'~-1 Iv 2 -

(~-1 I v1 -

i kv2 Ix) + (r'

i kv1 I)

x(r- 8I v 1) } dr 2 ::: c {2

;~~

!h(r)1 +

Ilvll1_(~,(n,oo)lIv2

- ik v 2 11 6_ 1 ,(n,oo)

+ Ilv, - ikVll1o_1,(n,oo)IIV2:I_o,(n,oo)}

On the other hand,

pnv j

tends to

v

j

in

H6:~o(I,X)

as

n-roo ,

and

hence (8.26) as

nroo.

(8.12) follows from (8.25) and (8.26).

general case that

Q,.

J

c

F~(I,X).

Let uS next consider the

Then there exist sequences

(j

(n=l,2, ... , j=1,2) , (8.27)

(j=l,2)

92

Let

v (j;1,2, n;l ,2, ... ) jn

be the radiative function for

{L,k,,Q,nj}.

Then we have from Theorem 2.3

v. -)- v .

(8.28)

as

n700

letting

Jn

for j=1,2. n700

J

(8.12) in the general case is easily obtained by

in the relation

(8.29) which has been proved already.

(8.13)~'(8.15)

directly follows from

(8.12) .

Q.E.D. {F(k)f/f E L ,c(I,X)} 2

Now it wi 11 be shown that the range D,

and hence it is dense in

defined by (5.36), i. e. ,

O~.:;(

{

for

xED

definition of W

o

)J(y,k)

r,(r)=O

( r; 1) ,

= l( r;2)

I

Set

r() i,l(r.,k) rex

~

Then, as can be easily seen from the

and (5.20) in Lemma 5.5,

is the radiative function for PROPOS IlION 8.5.

s(r) be a smooth function on

2 fO(r) = (L-k )w O

kE R - {a}.

and

and

r );1

wO(r) = (8.30)

Let

X.

contains

Let

f EL 2 ,cS(I,X) O

and

fL,k,tjf I L O

x E': D and 1et

f0

be as above.

Then

(8.31 ) PROOF.

Let

xn(r)

(O,n), n=l,2, ... , for

{L~k,,Q,IXnfOI}

be the characteristic function of the interval

and let and

w and n

'In

be the radiative functions

{L,k,.Q,[ (l-xn)fOJ} , respectively.

Note that

93

Xnf OE L2• 1+ (I.X), because the support of \/0 is compact. S the relation Wo = wn + fin and Theorem 5.3. we have

Then. by

(8.32) in

X.

(8.33)

Since

e

-h

IwO(r) = x for

r;2. there exists the limit

" xn = s - 11m e-ip(ro ,k) n () r

r-HlO

in X •

n

and hence the relation

x-x n

(8.34) is valid for each

n=1,2, ...

On the other hand, proceeding as in the

derivation of (8.4). we obtain from (3.32). (8.35)

Inn (r)

I

~

;

~ : Tl ~ ( r)

- i knn(r) : ~ +

-m-

II (1 - Xn )fOil

~

(n=1.2 .... ) with

C=C(k). whence follows by letting

{r}

1n (8.32) that

m

~xn

along a suitable sequence

(n-l.2 •... )

(8.36) Thus we arrive at (8.37)

I F( k )( X f ) - x I ; CII (1 - X ) f II x n 0 () n O '

(C = C(k). n=1.2, ... )

and (8.31) is obtained from (8.37).

Q.E.D.

94

In the remainder of this section we shall consider the restriction of

F(k)

'tihich is denoted by

to

L2 ,o(I.X) consider the mapping (8.38)

[

+

:J

F(k)

again.

Let

US

k -;- F( k)c F. ( L2.6 ( I , X) • X)

can be seen a B(L 2 .(;(I.X),X) - valued continuous function on R-{O}. Further we can see that F(k) is a By the use of Lemma 4.3

F(k)

compact operator from THEOREM 8.6. (I)

Then

into

X

Let Assumption 5.1 be satisfied.

F(k)

is a

B(L2.~(I.X),X)

-valued continuous function

on R-{O} . (II) For each L2 ,5(I.X) PROOF.

into

kE

R-~O}

F(k)

X

Let us assume that

(i) is not true at a point

Then there exist a positive number

IIf n" o

(8.39)

is a compact operator from

c

>

0 and sequences

k c R-{O:.

{f } n

C

L2• (I.X). 6

(n=1.2 •... )

(n·'CX) ) kn -+k IF(k n )f n - F(k)f n Ix

~

c

(n=1,2 •... )

With no loss of generality

{f } may be assumed to converge weakly in n L2 • o(I.X) He shall show that there exists

L2•.s(I.X) to some f E subsequence {n m} of positive integers such that {F(k

{F(k)f n } and m F(k)f as fll-+OO.

)f } converge strongly to the same limit nm nm which will contradicts the third relation of (8.39~ We shall consider

95

the sequence

[F(k)f} n

only, because the sequence

n

be treated in quite a similar way. k=k

n '

f=f

n •

v=v

n

[F(k)f n} can

It follows from (8.13) with

that (n=1.2 ... )

(8.40 )

vn is the radiative function for {L,kn.~lfn I}. By Lemma 4.3 a subsequence {v } of {v} can be chosen to satisfy {v } nm n nm converges in L ._ (I,X) to the radiative function v for {L,k,,Q,lfl}, 2 8 whence we obtain

where

(8.41 )

Let

xED

and set r( ) ijJ(ro,k n ) r e m x,

r wn (r)

(8.42)

where

r,(r)

I

m

Il

gn (r) m

~

(L-k 2n )w n (r) m m

is defined by (5.36).

Then it follows from Proposition 8.5

that (8.43)

F(k

n

)g

m

n

=

x

m

By taki ng note of Remark 5.9, (2) it can be eas i ly seen that

(8.44)

w --)- w nm 0

in

L2,_S(I,X)

g .,. f n 0 m

in

L2,6(I,X)

I

96

as

with wo and

nJ-¥X)

f O defined by (8.30).

Therefore we have,

using (8.13) in Theorem 8.4,

(8.45) 1

2ik {(v,fO)O - (f,wO)O}

=

= (F(k)f,

F(k)fO)x = (F(k)f, x)x '

which, together with (8.38) and the denseness of that

{F(k n )f n } converges to rn m

shown (i).

D in

F(k)f strongly in

X

X , implies Thus we hav2

In order to prove (ii) it is sufficient to show that

{F(k)f n} is relatively compact when L2 ,0~(I,X) proof of (i). in

This can

rf n} is a bounded sequence

be shown in quite a similar way used in

the

Q.E.D.

THEOREM8.? let

~ER-{O}.

Let Assumption 5.1 be satisfied.

Let

ff::L2,o(I,X) and

Then

(8.46)

F(k)f

k) s - lim e -i'l(r· t n ' v(r) n

in

X,

n,)<)O

where

v is the radiative function for

sequence such that

and

{r } is an n X as n-¥X).

vl(r ) - ikv(rn)-+O in n Before showing this theorem we need the following the Green

formula.

rnt oo and

[L,k,Q,[f]}

97

LEt4MA 8.8.

Let

V

j EL 2 (I,X)loc

satisfy the equation

(8.47)

(8.48 ) =

(v,(r) - ikvl(r), v 2 (r))x - (vl(r),vZ(r) - ikv 2 (r))x + 2ik (vl(r), v (r))x 2

for

rEI.

PROOF.

The idea of the proof resembles the one of the proof

of Proposition 3.4.

As has be shown in the proof of Proposition 1.3,

-v·-=-tj - 1v. E H (R)1 N 2

(j=l,2). Therefore we can proceed as in the oc proof of Proposition 3.4 to show that there exist sequences 't ln } and J

J

{4>2n}

such that

¢In,¢;2nE:C~(I,X) and ~.

-+ v· J

'j

n

rpjn(r) -+vj(r)

in

L2 (I,X)loc X (r cT)

:j:J'. (r) -+vl.(r)

in

X ( rE::I )

In

(8.49) Jn

(L-k 2 ):j:J. In

as

n-l-OO

(8.50)

(j=1,2).

J

f . -+ f. nJ J

Integrate the relations

in

L2 (I, X) 1oc

98

from

0 to

r and make use of partial integration.

Then it follows

that - (Qln(r), ¢2n(r))x + (~ln(r),Q'2n(r))x (8.51)

=

(f ln '¢2n)O,(O,r)

- (~ln,f2n)O,(O,r) (¢In(0)'~2n(0))x =

where we should note that

(Qln(0)'¢2n(0))

0 .

(8.48) is easily obtained by letting n·>«> in (8.51). Q. E. D.

PROOF of THEOREM 8.7.

Set in (3.32)

and

k =0

2

r

=

rn

Then we have (8.52)

1

2 v( rn) x ;;

which implies that (8.48)

1k2 v (r n ) - i v(r n )! x2 - £k1m (f, v)0 , (0 ,r ) ' n

1

vl = v2

=

I

1

{lv(rn)l x} is a bounded sequence. v,

f

l

=

f2

=

f

and

r

=

r

Next set in

Then it follows

n

that e -il.l(r . n• ,k) v( r n ) x I

1

(8.53) +

t Im(v(r n).

=

1 ( ) f 1m v.f O,(O,r ) n

vl(r n ) - ikv(rn))x

The second term of the right-hand side of (8.53) tends to zero as n·>= because

IV1(r n) - ikv(rn)lx'>-O as

n~)

and

Iv(rn)l x is

bounded, and hence we have (8.54)

1 i In

n-><x>

1

e - i lJ ( r n"' k ) v(r ) 2 = I

n x

1 k

I In (v f ) = ' 0

1

F( k) f 2 , 1

x

99

where (8.15) has been used. (8.30).

Let xED and let

Then, setting in (8.48)

vl=v.

o and

W

f O as in

v2=w • fl=f. f 2=fO and O

r=r n,

we obtain (e-i~(rn··k)v(r)

n •

(8.55)

e-i~(rn··k)w (r )) 0

n

x

= 2~k {(v(rn),wO(r n) - ikwO(rn))x - (Vi (rn)-ikv(r n), wo(rn))x

Since e-i11(rn",k)w (r) = x (r ~ 2) and wO(r ) - ikwO(r ) n n o n n it follows from (8.55) that -iZ(rn")wO(r n) -+0 in X (rn-+oo) (8.56)

11· In ( e -i~(r n ".k) V ( r ) • x )

n-+OO

- 1 {( f) n x - 2i k v. 0 0 = (F(k)f, F(k)f )x = (F(k)f.x) x • O

where (8.14) and Proposition 8.5 have been used.

From (8.54), (8.56)

and the denseness of D (8.46) is seen to be valid. Q. E. D.

CHAPTER III SPECTRAL REPRESENTATION The Green kernel

~9.

In this chapter the results obtained in the previous chapters will be combined to develop a spectral representation theory fOr the operator L=-

(9. 1 )

d

2

-~+

B(r) + C(r)

dr

given by Q(y)

(0.20)

and

(0.21).

Throughout this section the potential

is assumed to satisfy Assumption 2.1. Now we shall define the Green kernel

(r,s s

E

E:

f = [O.m) , k C

I • x

C

[+)

G(r.s,k)

and investigate its properties.

X and let lIS,x]

Let

be an anti-linear function on

Hl&B(I,X)

defined by (9.2)

= (x,¢(s))X

Then it follows from the estimate

(9.3) which is shown in Proposition 1.1. that l[s.x]

~

FS(I,X)

B~ 0

for any

and the estimate

(9.4)

is valid.

Denote by

{L.k.t[s.x]}.

v

=

v(·.k.s.x)

the radiative function for

Then, by the use of (1.8) in Proposition 1.3 and (2.7) in

Theorem 2.3. we can easily show that !v(r)i (9.5)

x ;' C(l

+ s)I:)IX1x

(C = C(H,k), r

C

10,RJ, s c I , x

C

X)

101

for any

REI.

Therefore a bounded operator

G(r,s,k)

on

X is well

defi ned by G(r,s,k)x

(9.6)

DEFINITION 9.1.

v(r,k,s,x).

(the Green kernel).

G(r,s,k)(r,s e T , k E t+)

will be called the Green kernel fOr

The linearity of the operator linearity of t[s,x]

The bounded operator

G(r,s,k)

with respect to

x.

L .

directly follows from the Roughly speaking,

G(r,s,k)

satisfies 2

(L - k ) G(r,s,k) ; 6(r - s) ,

(9.7)

the right-hand side denoting the Green kernel

-

on -

I

Then

x [

-

I x I x [

+ +

(i i )

x

Let Assumption 2.1 be satisfied.

G(·,s,k)x

X

Further,

is an

L2 ,_o(I,X)-valued continuous function

G(r,s,k)x

is an X-valued function on

x X , too.

G(O,r,k)

~

(r,k)

G(r,O,k) ; 0 for any pair

(iii) Let (s,k,x) E T

x [+ x

X and let

interval such that the closure of J have

The following properties of

G(r,s,k) will be made use of further on.

PROPOSITION 9.2. (i)

a-function.

J

I

x [

+

be an arbitrary open

contained in

G(·,s,k)x E D(J) , where the definition of

E

I - {s}. D(J)

Then we

is given after

the proof of Proposition 1.3. ( i v)

exists

C

(9.8)

where II II

Let

R > 0 and let

C(R,K)

K be a compact set of [ + .

such that

JI G( r. s , k) II

< C

means the operator norm.

(0

:s.

r, s

:s.

R, k

E

K) ,

Then there

102

(v)

We have for any triple

(r,s,k)

I

C

x

I

x

[+

G(r,s,k)* = G(s,r,-k) ,

(9.9)

G(r,s,k)*

denoting the adjoint

PROOF.

of

G(r,s,k)

Let us first show the continuity of i[s,x]

FG(I,X) , ,3 ~ 0 , with respect to

sET and

x co X.

in In fact we obtain

from Proposition 1.1.

o

~ l(x - Xl, (l+s)l-'rp(s))xl +

(9.10)

+ I(x ' , ((l+s)13 - (l+s,)l3)(p(s'))x l

~ {12l x - Xl: X(1+s ) P + Ix IX(1 +s );:s Is - S 1

whence follows the continuity. v

for

{L,k,R}

R E:

[

+

and

:;2

+ 11""2 Ix I I XI(1+s ) S - (1 +s I ) BI }H!I (',

By recalling that the radiative function

is continuous both in

with respect to

I

(C

L2,_~(I,X)

and in

HloG(I,X)lOC

F.(I,X) , (i) follows immediately. o

G('ls,k)x G H10,B(I,X)1 we have G(O,s,k)x = 0 for all oc , + (s,k,x) I x [ x X , which means that G(O,s,k) = O. On the other

Since

hand, <-f.LO,x],;?> G('IO,k)x G(r,O,k)x

=

(X,¢(O))X

0

~

0

Q E

H\)B(I,X) , and hence

~(r)

Therefore

by the uniqueness of the radiative function,

on

Let us show (iii). such that

sufficiently small neighborhood of

'" v(r)

for all

is the radiative function for {L,k,O}.

is completely proved. function

=

~)(r)G(r,s,k)x

Thus (ii)

Take a real-valued, smooth

y(r) = 1 on r = s.

J

and

0 in a

Then it is easy to see that

is the radiative function for

g(r) = -·i'(r)G(r,s,k)x - 2'LiJ' (r)dd G(r,s,k)x. r

~(r) =

{L,k,.l[gJ}

\'Jith

Proposition 1.3 can be

"'-

made use of to show that

v E 0(1) , which means that

G(',s,k)x G O(J)

103

(jv) is obvious from (9.5). v(t) =

= G(t,s,k)x

and

Let us enter into the proof of (v).

w(t) = G(t,r,-f)x

with

x,x l EX.

Set

Then, setting

~ wand .; v in the relations "'n .n

respectively, and combining them, we arrive at (9.12)

(x,w(s))X - (v(r),xl)x = ~

.;~(tH(V' (t),w(t))x -

(v(t),w l (t))X}dt (n = 1,2, ... ),

where 'Pn(t) = .;(n(t-t )) , .;(t) O lR such that cp(t) = 1 (t:C 0), satisfy fact that

to> max (r,s) 'P

I

n(t)

-+

Let

-dt-tO)

is a real-valued, smooth function on

n as

0 (t

=

> 1) , and to is taken to in (9.12) and take note of the

-r

·X)

n

-r -.~).

Then we have

(x,G(s,r,-k)x')X - (G(r,s,k)x,x')x (9.13)

(to> max(r-s)) . [3y rr,aking

to

-+

<Xl

along a suitable sequence

an}' the right-hand side

of (9.13) tends to zero, whence follows that (9.14) for any

(x,{G(s,r,-k) - G(r,s,k)*}x')x = 0 x,x l EX.

Thus we have proved (v) . Q. E. D.

104

310.

The eigenoperators

The purpose of this section is to construct the eigenoperator n(r,k)

(rET,kE::R-~O})

was defined in 39.

by the use of the Green kernel

G(r,s,k)

In this and the following sections

which

will

Q(y)

be assumed to satisfy Assumption 5.1 which enables us to applY the results of Chapter II. We shall first show some more properties of the Green kernel in addition to Proposition 9.2. PROPOSITION 10.1.

Let Assumption 5.1 be satisfied.

Then the following

estimate for the Green kernel + ( k'. [ , I rl k ;> 0, r , S :.- I)

C1( k) (10.1 )

/1 G( r , s 1 k ) /i

<

{

:>

C2(k)min~(1+r)

I

1+ 3

,(l+S)

1+

15

t ( k c R - "·0 :. 1 r, s r- T)

(O
holds , where and

C2 (k)

are bounded when

and R - {O},

respectively.

k

InOVPS

holds for any radiative function kE::(C+

and PROOF.

f

C

compact set in

. + ) {kE::[ /Tl1lk>O;

r G( r. s , k) f ( s )ds

(r E:: T)

J[

v(o,k,Q, If I )

for

{L,k,y.,lfl}

with

0 .

Then it follows from

L ,()(I,X). 2

(i) Assume that

Lemma 2.7 that

il

Further

v( r, k, Z If 1 ) =

(10.2)

in

v ~ G( ,s,k)x

k (~ [ (5

+

T,

wi th X

X)

IlIlk

~

belonqs to

116,[3(I,X) .

The

first estimate of (10.1) is obtained fron! (9.3), (2.16) in Lemma 2.7 and

(9.4) \'iith

;3=0

Next assume that

k'"-R - {Q~.

Applying (5.11)

10S

in Theorem (5.4) and using (9.4), we have which implies that IIG(rls,k)lI.;

(10.3)

C(l+s)l+i<'

(r,se!)

The second estimate of (10.1) follows from (10.3) and the relation G(r,s,k)* '" G(s,r,-k) ((v) of f)roposition 9.2). with compact support in U'"

to

k(,- [ + , Imk > 0

and let

G(",r,-k)x and the radiative function H6 1[3(I,X) and satisfy

(10.4)

(ii) Let

v for

fCL (I,X) 2

Then (L , k,9,1 f

I}

belong

{ Set ¢

Then , using the relation

v in the first relation of (10.4) .

=

G(s,r,-k)*: G(r,s,k),

we have

(10.5) '" (G(",r,-k)x,f)o = (xl!IG(rls,k)f(s)ds)x whence (10.2) follows.

Let

f

be as above and

k

~

R - :01.

Then we

ik + j.,. (n=l,?, .. ,) to obtain (10.2), n where we have made use of the continuity of the radiative function can approximate

v(",k,.q f])

k by

with respect to

k and (9.a) -in Proposition 9.2.

(10.2) has been established for support in

T.

In the case that

k lf c

[

+

and

f

L21~(I,X)

~

L?(I,X)

Thus

with comoact

we can approximate

fnE.. L2(I,X) with compact support in r Then, taking note of the continuity of the radiative function v(",k,x,1 f I)

by

{Fn}' where

with respect to

f

and the estio0te (10.1), we arrive at (10.2).

n... E. D.

f

lOG

Let

k

t

R - fO; , scI

x

v is the radiative function for with

C

X and set s~x

[L,k,tl

Illlls,xl 11!1+3 ~ ,17. (1+s)lH 1xlx '

I

~

v(r)

Q,ls~xl

and

\vhere

G(r,s,k)x .

=

c Fl+:-~(I,X)

is qiven by (5.9).

f<

Therefore Theorem 5.3 can be applied to show the existence of the limit (10.6)

F(k)~r

s-lirne -i"(r· , k) v(r)

s ,x I

in X .

r-:><>;

It follows from Lemma 8.1 that

Ir (k) £1 s ~ x1 I ;

(10.7) with

C = C(k)

CII ],1 s , x 1 II (~

and hence for each pair

~

'1( r ~ k)

bounded linear operator

=

~ 1"7. C(1+sf' Ix Ix

(r,k)<::T

x

a

X is \vell-defined by

on

s - lim e-i.,(t.,k)G(t,r~k)x ",r(r,k)x

(10.8)

:O~)

(R -

(x

(~

X)

Y'-+OC

DEFINITION 10.2.

r(r~k)

The bounded linear ooerator

(10.8) will be called the

~~~n0JP-~~~to~

defined by

L.

associated with

The appropriateness of this naminq will be justified in the remainder of this section (esoecially in Theorem 10.4). PROPOSITION 10.3.

Let Assumption 5.1 be satisfied.

s - 1i m G( r, s , - k)e i lJ ( S • , k)x

(10.9)

=-

Then we have

rt(r ~ k)

X

S4<Xl

for any triple (5.8) and PROOF.

(r~k~x) c

n*(r,k)

T

is the adjoint of

Suppose that there exist

3 >0

and a sequence 0 Ivn(r o ) - r] * (rO~ko)xolx

Vn (

(R - {ol)xx~

x

r

>

such that ~ ~O

Tl(r~k)

o

~I(y,k)

is defined by

.

kO ~ R - :01 s nt m and

0,

holds for all

r) = r.( k ) i~(s n ·~k) X ~ r,s~- 0 e

where

n=1~2~ ... ~

~ X

oE X~

where we set

By the use of the interior estimate

107

(Proposition 1.2) and (10.1) it can be seen that the sequence [!I vnJI B,(O,R)}

is bounded for each

R>0.

Rellich theorem) there exists a subsequence

Gy Theorem 1.5 (the

fWpl of {v n } such that

f Wp} is a Cauchy sequence of

estimate again.

Then

L2(I,X)loc. Make use of the interior {w p} is seen to be a Caucy sequence in

Therefore it follows from (1.8) that ~wD(rO)} is a . , Cauc hy sequence ln X. S·lnce G( rO,s,-k ) e ill(so,k) Xo converges O to n*(rO,kO)x o weakly in X by (10.8), wp(r o ) converges H6,G(I,X)lOC·

to

r,*(ro,ko)x O strongly in

X, which is a contradiction.

Let us summarize these results in the following THEOREM 10.4. ( i)

Let Assumotion 5.1 be satisfied.

Then s - lim e- i11 (so,k)G(s,r,k)x

dr,k)x

(10.10)

in X ,

s K"

and r.*(r,k)x

(10.11)

s - lim G(r,s,_k)ei:J(So,k)x

in X

s-~·'"

for any triple (1o. 12) where

(r,k,x) ("" I!

II

!I

'1 ( r,

r

x (R -

k) ~ ~ :::

fO}

~~e

)xX

I: '1 *(r, k) II

means the operator norm and

<

have

C(1+r )(' , C;;; C(k)

is bounded when

k moves in a compact set in R - fO} . (1i) (10.13)

The relation 2ik(n(s,k)x, = ({G(r,s,k) - G(r,s,-k)ix,x')x

108

holds for any

X,XI(~X,

kf.:R - (OJ

and

1 r*(0,k)XEH ,G(I,X)1 'iO(I) oc o,

(iii)

rc-T, kCR -

PROOF.

and satifies the equation

2 (L - k )v(r) = 0

(10.14) where

r,se-T.

{O} , xE::X .

(i) follows from (10.8),

in (8.12)

£.1= £Is,xl ';2

Then, notinq that

Proposition 11.3 and (10.7).

r,fr,x'l ,

=

Hk)£ls, xl '"

T~(s,k)x

v1:oG(0,s,k)x and

and

F(k)X,[r,xl

v =

2

Set G(o,r,k)x

=

r(r,k)x ' ,

we obtain,

2-h {d,1 r,x' 1 ,GT~---;k)x)

(r;(s,k)x , r-(r,k)x)x (10.15)

- (9,[

2~k

vs(r)

0

,

r, k) X I)

Let us show (iii).

G(s,r,k)* = G(r,s,-k), Set

vs(r)

==

completes the i'l(So k) G(r,s,-k)e I , X.

satisfies 00

(10.16) Let

}

{ (G(r,s,k)x,x')x - (x,G(s,r,k)x')x }

which, together with the relation proof of (ii).

s, xl, G(

IS

(CD

n

} , sn e- 1 , be an arbitrary sequence such that

co Co (l,X)) . s tex> n

as

n-+O:>

Then, as we have seen in the proof of Proposition 10.3, there exists a subsequence L (I,X) . 2

L2 (r'X)loc

~tp}

This means Letting

of

{s} n

{v }

s

s)-<Xl

} is a Cauchy sequence in tp itself converges to :;*(o,k)x in such that

{v

in (10.16), we arrive at

(10.17) Proposition 1.3 can be applied to show that

~*(o,k)x E H~,G(1,X)10C Q. E. O.

n 0(1)

109

The following two theorems will show some relations between the eigenoperator and the operator

F(k)

THEOREM 10.5.

Let Assumption 5.1 be satisfied.

be as in

Then we have

§8.

rR

F(k)f = s-lim:

(10.18)

R-;>
Let F(k) ,

kE

R - ~O}

in X

~(r,k)f(r)dr

In particular

for any F (k)f

(10.19)

=r J

holds for

ff:L 2o (I,X) , p

I

with

p(r,k)f(r)dr b

>

-2t + (~.

where the integral is absolutely

convergent. PROOF.

,

Let

f c L2'o(I,X) and define fR(r) by (0 :s r < RL (f(r) fR(r) '" L o (r > R) • ;;

(10.20)

Then it follows from Theorem 5.3 and (10.2) that F(k)fR = s - 1i m e- i 11 ( r· , k) v(r k r-+<x)

"

Q,f f.

R

I)

= s - lim e-i~(r.,k)fl G(r,s,k)fR(s)ds r-)
(10.21)

R '" Jr s _ lim

o

'" JR

o

{e-i~(r.,k)G(r,s,k)f(S)}

ds

r-+<X)

~(s,k)f(s)ds ,

where we should note that we obtain from (10.3) IG(r,s,k)f(s) Ix

(10.22)

; C(k) (l+s) 1+13!f(S) Ix

and hence the dominated convergence theorem can be applied. converges to

f

in

L ,6(I.X) 2

as

R->o:: and

F(k)

Since

fR

is a bounded linear

110

L2,~(I,X)

operator from n-h»

in (10.21).

into

X,

(10.18)

is obtained by letting

"'I,X) with r > ~ + 0 , then it follows 2 h(s,k)f(s)!x is integrable over Therefore we

If

fCL

, f)

from (10.12)

that

obtain (10.19) from (10.18) . Q. E. D.

THEOREM 10.6. (i) XE

Let

Let Assurr1ption 5.1 be satisfied. bOoR - {Ol

Then

r·*(·,k)x

C

L2,_6(I,X)

for any

X with the estimate n n* (•, k ) xii,.

(10.23) C ~ C(k)

where

(i i)

CIx I

<

-0

(x ex)

x

k moves in a compact set in R - :01 .

is bounded when

He have

for any triple

(k,x,f) '= (R - '.0;' )x X x L?(J,X)

PROOF. (i) Let 9

C

L2(I,X)

with compact support in 1

Then we obtain

from Theorem 10.5 (

(10.25)

(F(k)g,x)x

~ (I

JI

l(r,k)g(r)dr, x)

JI (g(r), Set in (10.25)

x

n*(r,k)x) dr x

g(r) ~ XR(r)(1+r)-28~*(r,k)x, \R(r)

function of (O,R) L2 ,8(I,X) into X,

Then, noting that we arrive at

F(k)

being the characteristic

is a bounded operator from

111

R

1r ()-25 l+r In*(r,k)x 12xdr

= (F(k)[xR(l+r) -2'5 -'n*(",k)x},x)x

0 <

(10.26)

C(k)

"XR(1+r)-28n*(",k)XH~ Ixlx

= C(k) IIn*(o,k)xll_~~,(O,R)lxlx

'

which implies that (10.27)

Since

R >0

(ii) Let R-rcn

is arbitrary, (10.23) directly follows from (10.26).

f C L ,o(I.X). 2

Set in (10.25)

Then, since (f(r),

g(r) = xR(r)f(r)

and let

~*(r,k)x)

is integrable over x (10.23), we obtain (10.24), which completes the proof. •

by

Q. E. D. In order to show the continuity of r:(r,k) and respect to

Let

T'.*(r,k)x;

(10.28 )

x

(=

D.

Then we have

o J (r x ,k) - h( r" k x ) 2i1 k I.,r() r il e (r c:

~(r)

(r ~ 1) ,

=

I,

k

C

is a real-valued smooth function on 1 (r ~ 2),

radiative function for (10.29)

with

k we shall show

PROPOSITION 10.7.

where

q*(r,k)

~(y,k)

R - {Q}),

I

is as in (5.8) and

{L, -k, £1 f I}

with

such that ~(r) h(",k,x)

=

is the

0

112

PROOF. Then

= ~(r)e

Let wO(r) f OE::L 2,8(I,X)

f L, k, £1 f O]}.

i~(r'

' k) x and

2)w

O as in (8.30). and Wo is the radiative function for

gee ~ (I , X).

Let

fO~(L-k

and making use of the fact that

The n, set tin gin (8. 11 ) r(k)fO'=x

f 1= f 0 ' f 2= g

by Proposition 8.5, we have

(l 0.30)

with the radiative function for

{L,k,Q.[gl}.

By (ii) of Theorem 10.6 the left-hand side of (10.30) can be rewritten as (10.31)

(x,F(k)g)x

=

(n*(·,k)x. g)O

.

As for the second term of the right-hand side we have, by exchanging the order of integration, ,.0)

(fO'V)O

=

reX}

jO(fO(r), JOG(r,s,k)9(S)dS)x dr

(10.32)

(2ikh('),9)0

'

where we have used (10.2) repeatedly.

(10.30) - (10.32) are combined

to give (10.33)

(r~*(.,k)x,g)O = (

1

ilk Wo - h, g)o

.

(10.29) follows from (10.33) because of the arbitrariness of 9 E C~(I,X) Q. E. D.

113

THEOREM 10.8. ~*(r,k)x

PROOF.

Let Assumption 5.1 be staisfied.

are strongly continuous Let us first show

Then

n(r,k)x

X-valued functions on

the continuity of Tl*(r,k)x.

Ix (R - {OJ ) xX. To this end

it is sufficient to show that n*(r n,k n)x n tends to n*(r,k)x in X , where r n -+r in T , kn -+k in R - ~O}, xn -)- x in Let

€

>

0 given

Then, by the denseness of

(10.12), there exists

X

o

C

{

for

Therefore we have only to show that

n-}<X)

nO

such that

in X

(10.35)

for

X as

~ ,

(10.34)

n; nO'

strongly

D and the estimate

D and a positive integer

In*(r,k)x - n*(r,k)xol <

and

xED.

In fact it follows from Proposition 10.7 that n*(r,k)x - r.*(rn,kn)x =

(10.36)

2~ k IF, ( r) ei 11 ( r 0, k) - ~ (r n) ei iJ ( r nO, kn ) }X

+ fh(rn,k,x) - h(r,k,x)} +

Since

)J

(y, k)

definition of

{h(r ,k ,x) - h(r ,k,x)} n n n

is conti nuous in 11 (y , k)

(y, k)

by (1) of Rema rk 5.9 and the

(s ee (5.8)), the fi rst term of the right-hand

side of (10.36) tends to zero as

n,(X) .

The second term of the right-hand

si de of (10.36) tends to zero because of the continuity of the radiative It follows from (1) of Remark 5.9 that (L-k~) (~eiv(ro,kn)x) converges to (L-k 2) ((ei).(ro,k)x) in L2,5(I,X) ,

function

h(r,k,x).

114

which implies that h(·,kn,x) L2,_a(I,X) n H6,B(I,X)lOC

converges to

Since the convergence in

means the uniform convergence in [O,R] on

I,

h(·,k,x)

in

H6'~(I,X)lOC

X on every finite interval

we have

o

s - 1im

(10.37)

in

X •

n-)o
Thus

(10.35)

has be proved.

The proof of the continuity of

r.(r,k)x

is much easier.

In fact, the continuity follows from the facts that

(10.38)

n(r,k)x

~ F(k)~r,x]

(roE!, kER - {OJ, xE:X) ,

and that as has been seen in the prooF of Proposition 9.2, X,[r,x] is an

F~-valued, ()

continuous function on

YxX

and that

B(Fo(I,X),X)-valued, continuous function on R -

F(k)

is a

{O~.

Q. E. D.

115

§ll.

Expansion Theorem

Now we are in a position to show a spectral representation theorem or an expansion theorem for the Schrodinger operator with a long-range potential.

To this end the self-adjoint realization of T -

-~

+ Q(y)

should be defined and some of its spectral properties should be mentioned. Here let us note that these properties were never used in the previous sections (§l - sla). be a real-valued, bounded and continuous function on lR N Then two symmetric operators h and h in L (lR N) are defined by 2 a Let 1(y)

D (h)

a

~

p (h)

C'aX> (

lR N)

h f ;;; T f = -.::, f

(11.1)

a

~

hf

a

=

Tf

=

As is \<Jell-known,

-i\f + Q(y)f

h is essentially self-adjoint in a unique self-adjoint extension H defined by a

L (lR N) with a 2

(11.2)

where the differential operator tion sense. (11.3)

T should be considered a Further, it is also well-known that we have

lTl

the distribu-

116

Here

o(H O)

is the spectrum of

HO and

Gac(H O)

is the absolutely

continuous spectrum. THEOREM 11.1. lR

N

with

Q(y)

-r

0 as

Iyl

H.

(11.4)

\~e

-}-

l'..'()

Let

ill

L (lR N)

essentially self-adjoint in extension

be a real-valued, continuous function on

Let Q(y)

2

h be as in (11.1).

Then

h is

with its unique self-adjoint

have

r D(H)

;;; V(H O)

1

If =

Hf

=

-6.

+ Q(y)f

(in the distribution sense).

kO ~ 0 and Ge(H), the essen[0,"") Therefore on [kO'O) the

H is bounded below with the lower bound tial spectrum of H , is continuous spectrum of

equal to

H is absent, and the negative eigenvalues, if

they exist, are of finite multiplicity and are discrete in the sense that they form an isolated set having no limit point other than the origin

0

PROOF.

There exists no positive eigenvalue of H . Let

V;;; Q(y)x

be a multiplication operator Q(y)

Vis a bounded opera tor on (11.5)

L ( lR N) 2

h

= hO +

and

Then

his rewritten as

V.

Since

hO is essentially self-adjoint, h is also essentially selfadjoint with a unique self-adjoint extenstion H = HO + V and the domain V(H) is equal to [J(H ) (see, e.g., Kato [1], p. 288, Theorem O 4.4). V can be seen to be HO-compact, i.e., if {f n} C D(H O) and {HOf n} are bounded sequence in relatively compact in

L2(lR N) ,then the sequence {Vf n} is In fact this can be easily shown by

the interior estimate and the Rellich theorem.

Therefore the essential

117

H = H + V is equal to the essential spectrum O (Je(H O) of HO (see Kato [1], p. 244, Theorem 5.35). Finally we shall show the non-existence of the positive eigenvalues of H N Suppose that I, > 0 is an eigenvalue of H and let '(J E H2(lR ) be the eigenfunction associated with A. Then v = U'{J is the radiative spectrum a (H) e

function for

of

{L,~,O}

radiative function.

and hence

v

=0

by the uniqueness of the

This is a contradiction.

Thus the non-existence

of positive eigenvalues of H has been proved. Q. LD.

Set

(11.6)

M= UHU* ,

MO and M are self-adjoint operators in L2(I,X) with the same domain UH 2 (lR'~) and are unitarily equivalent to HO and H , respectively. Let E( . j M) be the spectral measure associated with M. We shall be mainly interested in the

where

U is defined by (0.19).

structure of the spectrum of M on (-"',D)

(O,m) , because the spectrum on

is discrete.

PROPOSITION 11.2. compact interval in

Let Assumption 5.1 be satisfied. (0,00)

and let

f,g

€

L2 ,8(I,X).

Let J Then

be a

118

( E(J j M) f ~ 9)0

=

=

i~f;;; {k > O/k 2 PROOF.

J}

2k2

fIT and

zEit - lR

and

f

I mIZ> - 0

E

1T

F(k)

is given by Definition 8.2.

and

{L)IZ~£[f]}

M by

Let us note that L 2

(I~X).

. r(-k)g)Xdk ~

--- (F(-k)f~

We denote the resolvent of

R(zjM) ;;; (M - Zfl.

with

E:

H k) 9 )Xd k

TI

(11.7)

where

fn _~2 (F ( k )f ~

R(zjM)

R(zjM)f::

~

i.e.)

v(·~IZ).f[f]) for

IZ is the square root of

Here

Z

)

v(· )1Z)'([fj)

is the radiative function for

In fact this follows from the uniqueness of the radiative

R(zj~1)f c UH 2 ( lR N)

function and the fact that

.

Moreover let us note

that

(11.8)

lim R(k b+O

2

±

ib; M)f

=

v(·~

(k

:kl~

1

Eo.

R[f]) in L ._ (I)X) 2 ' cS

lR -LO}~ f (:. L jI)X)) ~ 2~ 0

which follows from the continuity of the radiative function with respect to

kelt

+

Then from the 1tJell-known formula ( E( J ; i~ ) f) 9 )0

(11.9)

;;;

---2~·

lim

IT' b-l-O

=

fJ {(R(A + ib;M)f,g)O -

(f~R(a

+ ib;M)g)O}da

2~·

lim I( {(f~R(a - ib;M)g)O - (R(a - ibjM)f)g)O}da ", b+O 'J

it fo 11 ows tha t (E(JjM)f)g)O (11.10)

;;;

2~riL{(v('~Ia)f[fl),9)o

=

2~if}(f'V(.,-;a-~t[gj))o

- (f,V(',laJI9J))O}da -

(v(·~-;a~{lf]))9)O}da

.

By the use of (8.14) in Theorem 8.4 (11.7) follows from (11.10). Q. E. D.

Let us set 2

(11.11)

F± (k)

COROLLARY 11.3.

(i)

=

+1 -:;;:

ikF(~k)

E((O,X»;M)M

(k

>

0) .

is an absolutely continuous

operator, i.e.,

(ii)

F± (·)f

(11.13)

PROOF.

L (0,oo),X,dk) fOr f C L ,s(I,X) with the estimate 2 2 oo 2 2 2 IF... (k)f! dk = 11 E((O,'~»);M)f11 ~ II fll . o x 00 E

f

Obvious from (11.7) and the denseness of

L ,6(I,X) 2

in

L (I,X) . 2, Q. E. D.

(11.14)

'-F:rf)(k) = 1.i.m. F±(k)fR R-)-

in

L ((0,<x),X,dk) 2

,)0

for

f E L (I,X) , where fR(r) = AR(r)f(r) 2

istic function of the interval

and

F

±

are bounded linear oper-

L (I,X) into L ( (O,.·;o),X,dk) F±* 2 2 operators from L ((0,oo),X,dk) into L (I,X). 2 2 ators from

F(k)

and

n(r,k)

Let us set

2

(' n± (r,k) (11.15)

*

=

is the character-

(O,R).

It follows from Corollary 11.3 that

tion between

XR(r)

~/1T ikn (r,~k)

2

n± (r, k) = +1 a

i kIl *( r , ~ k)

denote the adjoint By the use of the rela-

120

fo r

reT, k > 0 . PROPOSITION 11.5.

Let

Let Assumption 5.1, (5.4) and (5.5) be satisfied.

be as above.

F± (i )

Then

F

(11.16) (F /)(k)

±

and

F

±

are represented as fo 11 o\vs:

1. i .111. fR'l (r,k)f(r)dr R-+ x 'i

R r * (11.17) (F/)(k) 1* (r,k)F(k)dk '" 1.i.m. R-+m JR- 1 t f

where

(i i )

rR r J r , k) f ( r) dr

(F + f) (k) :;; s - 1 i m.

(11.18)

-

L (I,X) 2

in

L (I , X) and Fe L ( (0,,';) ,X ,dk) 2 2 Then fo r ea ch Let f to L , o(I,X) 2

E

L ((0,<:),X,dk) 2

in

)°

:;;

I

,

k> 0

in

X.

R r .,: ; 0

In particular, if

f E L ,s(I,X) 2

with

2>

t+ ~

, the integral of the

right-hand side of (11.18) is absolutely convergent. PROOF.

Let

of

(O,R).

xR(r)

F (k)f ±

fR(r) '" AR(r)f(r)

R

with the characteristic function

Then it fo11o'tJS from (10.19) in Theorem 10.5 that

2" ? :;; +/-:1 ikF(+ik)f:;; +/:;;:ik -

(11.19) =

f:

-

-

r )1

'i(r,_+k)fR(r)dr

".(r,k)f(r)dr ,

which, together with the definition of

rtf ((11.14)), proves (11.16) .

In order to prove (11.17) it suffices to show (11.20) for

Fe L2((0,'~J),X,dk) with compact support in (0,0..') , because

are bounded linear operators from Denoting the inner product of

L ((0,..XJ),X,dk) 2

L ((0,m), X,dk) 2

by

into

F±*

L2 (I,X) )0

again, we

121

have for any

fC c~ (I,X) . (F±F,f)O = (F, ~f)0

= f~ (F(k) , ICn+(r,k)f(r)dr)x dk

o '" f:~1

0 -

rf,c'

l :(r,k)F(k)dk , f(r) Xdr 0-0 -

r,<Xl

=

n: ( . , k) F( k)dk

-) 0

which implies (11.17).

f~o n*(r,k)F(k)dk

and

±

-

~

flo ' )

Here it should be noted that the integrals

r~

)0

n (r,k)f(r)dr are continuous in

r and in

k,

±

respectively, because of Theorem 10.8.

(ii) directly follows from

Theorem 10.5 and Definition 11.4. Q. E. D.

Thus we arrive at THEOREM 11.6 5.1 be satisfied.

(Spectral Representation Theorem). Let

(11.21) where

B be an arbitrary Borel set in E(B;M)

X

(O,(X»).

Then

= F~* XT7'."""" F+ , - v [3 -

is the characteristic function of

v'Ef

Let Assumption

I1f '" {k > 0/k

2

E

B} .

In particular we have (11.22) PROOF. J

in

It suffices to show (11.21) when

(0,00)

From (11.7) it follows (E(J;M)f,g)O

(11.23)

fOr

f,g

=

t~at

fIJ ((F+f)(k), -

(F+9)(k))X dk -

(F+f)('), (F+g)('))O IJ * = (F± X F+f~g)O IJL2 .,(I,X) , which implies that ,0 (X

E

B is a compact interval

122

(11. 24 ) Since

E(J;M)f E(J;M)

and

L (l,X) and 2 (11 .24) .

F

=

F* *x

X

L ,o(l,X) 2

(f t.: L o(l,X) 2 ,0

±

are bounded linear operators on

t+

I'J -

±

r f

I""J

±

is dense in

L (I,X) , (11.21) follows from 2

Q. E. D.

In order to discuss the orthogonality of the generalized Fourier trans forms

F

±

PROPOSITION 11.7.

Let

be the generalized Fourier transfOrms

F±

M.

associated with (i)

r ± wi 11 be sho\'Jn.

,some properti es of

Then

(11.25)

F E(B;M) = X ±

/"±,

18

(11.26) (11.27)

E( [3 ,• M) F+* = F+*(/.

(11. 28)

E( ( 0, co ) ; M) F./ = t ±'* ,

where

-

B and (ii)

X

"18

F± L (l,X) 2

- 113,

are as in Theorem 11.6. are closed linear subspace in

(iii) The following three condition

L ((O,ao),X,dk) . 2 (a), (b), (c) about F+ (F)

are equivalent: maps

E((O,<x:);M)L (l,X) onto L ((O,·x),X,dk). 2 2 * ~ 1) , where 1 means the identity operator. 1 (F_F_

(a)

F+(F)

(b)

F F *~ + + The null space of

(c)

PROOF.

consists only of

F+ (f_)

(i) Let us show (11.25).

Let

f

E

2 K = II F± E([3; M) f -

j;

F+ fll

/13 -

0

L (I,X) 2

0 Then

123

2

(11.29) = /I F+E(B;M)fll -

0

2 + /I X

F fll

IB ~

0

From (11.21) and the relations E(B;M)2

(11. 30)

E((O,w);M)E(B;M) = E(B;M)

we can easily see that J the proof of (11.25).

= J = J ' and hence K = 0 , which completes l 2 3 (11.26)-(11.28) immediately follows from (11.25).

(ii) Assume that the sequence

L (I,X) , be a Cauchy 2 sequence in L ((O,ro),X,dk). By taking account of (11.26) f n may be 2 * assumed to belong to E((0,oo);M)L (I,X). Then {F/+f n} is a Cauchy 2 sequence, since F+* is a bounded operator. Thus, by noting that

* F+F+f n

= E((O,~);M)fn =

{F f } , f + n

n

E

f n ' the sequence

{f n} itself is a Cauchy sequence

in

L (I,X) with the limit f c. L (I,X) Therefore the sequence 2 2 {F+f } has the limit F+f, which means the closedness of F+L 2(I,X) n in L ((O,oo),X,dk). The closedness of F_L 2(I,X) can be proved quite 2 in the same way. (iii) By the use of (i) and (ii) the equivalence of (a), (b), (c) can be shown in an elementary and usual way, and hence the proof will be left to the readers. Q. E. D.

When the generalized Fourier transform

F+ or

F

satisfies one

of the conditions (a), (b), (c) in Proposition 11.7, it is called orthogonal-.

The notion of the orthogonal ity is important in scattering

theory. THEOREM 11.8 (orthogonality of F±). and (5.5) be satisfied. orthogonal.

Let Assumptions 5.1, (5.4),

Then the generalized Fourier transforms are

124

PROOF.

vle shall show that

Suppose that

* F+F

to show that

F; O.

using

(11.27)~

=

F±

satisfy (c) in Proposition 11.7.

F E L ( (O~(:o),X,dk). vle have only 2 8 = (a2~b2) with 0 < a < b < w . Then~

0 with some Let

E(8;M)F * F = 0

we obtain from

+

F*v F=O, "18

(11.31)

which, together with (11.17), gives b

f

(11.32)

*

~+(r,k)F(k)dk

; 0

a

Since

a and

arbitrarily~

b can be taken

exists a null set e

(O~m)

in

* r ~ k) F(k) ; 0 (11 . 33) n+(

0

r

11

it follows that there

such that

* (r , k) F( k) ; 0

(r «'

noting that

Let

F(k)f

O

x ~

E:

D and take

k

n* (r,k)F(k)

where we have made use of the continuity of (Theorem 10.8).

T,

+

fO(r)

E

in

as in (8.30).

e) ,

r

C

Then~

x by Proposition 8.5 and using (ii) of Theorem

10.6, we have

(11. 34 ) (k if. e~ x

whence follows that

F(k);;: 0 for almost all The orthogonality of

k

r

C

(O,m)

D)

E

i.e.~

F

~

0

can be proved quite in

the same way. Q. LD.

Finally we shall translate the expansion for the operator

theorem~

Theorem 11.6

M into the case of the Schr~dinger operator H

us define the generalized Fourier transforms

F,

associated with

Let H by

125

F+ -

(11.35)

N-l 2 Uk = k--x

where

L2 ((Oloo),X,dk

11:( r, k), r

l

k

FU

+

L (IR N 2 are bounded linear operators from

If the bounded operators

(=

'

is a unitary operator from F+

L2 (IR N,dC).

U-

I, k > 0

I

~*(

)

on

onto

dE.:) L (IR 2

N Idy)

into

f1: (r,k) and

X are defined by

(11 . 36 ) 1±

r, k

r

-(N-l)/2 k-(N-l)/2 *(

q+ r I

k)

~

then

~'Je

ha ve R ~ 1• i . m. ( 11+ ( r, k) ¢ ( r· ) ) (w' ) dr R-+oo 0 -

f

(F+tP)(;,)

(11 .37)

.

~*

(F 'f)(y) + ~*

where

F+

(R ~*

.

1• 1 • m. , ( n+ ( r I k )f ( k· ) )(~;) dk R-+(;V ) -1 R

are the adjoint of

~

F+

and

y = r~1 ~ = kw'.

be the spectral measure associated with H.

Let

E(·;H)

Then we have from Theorems

11.6 and 11.8.

THEOREM 11.9. [3

be an

Let Assumption 5.1, (5.4) and (5.5) be satisfied. Let

arbitrary Borel set in (O,m)

transforms

F+,

L ( IR N,di:) 2

and

defined as above, map

Then the generalized Fourier E((O,..:c);H)L (IR N,dy)

onto

2

(11 .38) ~

where

X

IB

is the characteristic funct ion of the set

{r~ElRN/I[,12EB}

{(

t:

IR N/

126

§12.

The General Short-Range Case

In this section we shall consider the case that the short-range potential

Q is a general short-range potential, i.e., l Ql(Y) = o( !yl-l-t-:). Throughout this section we shall assume ASSUMPTION 12.1. (Q) , (Ql)

(~)

The potential

Q(y) = ~(y) + Ql(Y)

Further, 2.0 (Y)

in Assumption 2.1.

satisfies

is assumed to satisfy

in Theorem 5.1. Then all the results of Chapter I and 99 of Chapter III are valid

in the present case.

Therefore the limiting absorption principle holds

good and the Green kernel

G(r,s,k)

is well-defined.

In order to make

use of the results of Chapter II and §10 - §ll of Chapter III, we shall approximate 2. (y) by a sequence {Qln(Y)} which satisfies 1 I2.1 n (y) I = a ( I y i- L.l) ( n=: 1, 2 with a constant p

•• )

To this end let us define Qln(Y)

(n=l ,2, ... ) ,

(12.1) where :o(t) ~(t)

by

is a real-valued continuous function on

= 1 (t S 0) , - 0 (t

~

lR

such that

1) , and let us set

(12.2)

Since the support of Qln(Y) sections can be applied to

is compact, all the result in the preceding Ln.

According to Definition 8.3,

i27

Fn(k) , R E lR -

{a} ,

is well-defined as a bounded linear operator

F (I,X) into X The eigenoperator n(r,k)(r E T , k E lR - {a}) 6 is also well-defined. Here we should note that the stationary modifier

from

},(y,k)

and its

kernel

z(y,k) are common for all

because the long-range potential

QO(Y)

all the properties obtained in PROPOSITION 12.2.

(i) n=l ,2,. . .

into

F(k)

and that

F (k) n

X.

satisfies

Let

be the bounded linear operator from

X defined as above.

Then the operator norm II Fn(k)11 and

into

is uni formly bounded when

k moves ina compact set in

lR - {O}

For each F(k)

from

X such that

(12.3) in

F(k)

L is

§8.

k E lR-{O} , there exists a bounded 1inear operator Fo(I,X)

n .

associated with

Let Assumption 12.1 be satisfied.

Fn(k)(k E lR-{O}, n=1,2, ... ) F (I,X) 6

n

is independent of

We shall first show that an operator well-defined as the strong limit of

L , n=1,2, ... ,

F(k)£ - s - lim Fn(k)( n --)- 00

II F( k)11

(ii) F(k)

is bounded satisfies

~'Jhen

k moves in a compact set

(8.12) fOr any

(1'£2

E

in

lR -{O}

Fo(I,X) , i.e., we

have (12.4) being the radiative function fOr are also satisfied by (iii) Let

J

(8.13)-(8.15)

F(k)

xED and

let

fO(r) = (L - k2)(~ei~(r·,k)x) defined by (5.8).

{L,k,v.} .

Then

fO(r) with

be as in

(8.30), i.e.,

~(r) defined by (5.36) and ~(y,k)

128

(12. 5)

r( k)f

where we set

F(k)l[fOl = f"(k)f

(iv) As an operator from

°

O

"X

,

.

L2 ,r:;(I,X)

into

lI3 (L2,{"~(I,x),X)-valul:'d conLinuous function on

each

k

C

(v)

IR- {OJ Let

L2,":~(I,X)

E::::

E:

and let -ijJ(r' k)

n'

n -+ 'x>

v

k

PROOF.

rnA",

and

Further, fOr

L ,()(I,X) 2

in

into

X

Then

X,

n

{L,k,nfl:

v'(r ) - ikv(r ) n n

>

and

° in

{r n } X as

is a n

>- w

Proceeding as in Ule proof of Theorem 8.4 we obtain

(1 (k)f, F (k).f)

r

n

(12.7)

IR- to}.

IR- ·,01.

C

v(r)

is the radiative function fOr

sequence such that

is a

is a compact operator fronl

F(k)f = s - lim

(12.6) where

f

F(k)

X , F(k)

1

x

-"

= -.-- f<.r v > -<.£ v > + ({c - c Jv ,v )oj 1n 1p n p 2, k ' n ' p for

('

~-

r->(I,X) and

function for

n,p=l ,2, ... ,

{Ln,k,t} (:Lp,k.f 1

of Theorem 4.5.

~... here

Set

v (v ) n p

p

=<

n

is the radiative

in (12.7) and make use

Then we have

(12.8) with

C = C(k)

which is bounded when

IR - {OJ , where

II

11

" 8 ,-0

k moves in a compact set

is the norm of

H1 , 8, (I ,X) 0 ,- 0

in

On the other

hand we obtain from (12.7)

as

n,p

-+ "" ,

because by Theorem 4.5 both

vn

and

vp

converge to the

129

radiative function

v

fOr

in

{L,k,£}

,(I, X)

L,)

<:.,-0

satisfies IIC

1n

(r) - C (r)il = l,o(r-n) - .;(r-p)j ,!C (r)11 < c (l + r)-c 1 (r 1p O 1

E

f)

(12.9) II C1n( r) - C1p( r ) Ii

"r

0

( n, p

F(k)t

Therefore the strong limit have the estimate follows.

=:

-7

,.,.,)

•

s-lim F (k) n -r ·x, n

IIF(k)11 '£ C , \'1here

exists in

X and we

C is as in (12.8), whence (i)

(12.4) immediately fo11o\'1s by letting

n")-

co

in the relation

(12.11) v .,j=1,2, nJ

being the radiative function for

being made use of.

Thus

{L ,k,f..: J

11

(ii) has been proved.

and Theorem 4.5

Let us shO\'1 (iii).

Set

(n=1,2, ... ) .

(12.12) Then it follows from Proposition 8.5 that (12.13)

(n::ol,2, ... )

Fn( k)fOn ( r) = x

Further, it is easy to see from (5.20) in Lemma 5.5 that (12.14)

f On

-+

(n

f0

(12.5) is obtained by letting (12.14)

")-

"")

n -).

in OJ

L2 ,6(I,X)

in (12.13) and using (i) and

Re-examining the proof of Theorems 8.6 and 8.7, we can see that

(iv) and (v) are obtained by proceeding as in the proof of Theorems 8.6 and 8.7, respectively.

This completes the proof. Q. E. D.

Let us now define the eigenoperator (12.15) r;(r,k)x = F(k)R[r,x}

n(r,k)

associated with

(rE::T, kE IR - {O}, xCX),

L by

130

where :; E

<..'lr,x]

is as in (9.2), i.e., = (x,,>(r))X

H16B(I,X)

proposition,

At the same time, as will be shown in the following n(r,k)

~n(r,k)

operator

can be defined as the strong limit of the eigen-

associated with

PROPOSITION 12.3.

Ln·

Let Assumption 12.1 be satisfied.

be the eigenoperator associated with Then for each

(i)

C

(r,k)

E::

~

C(k)

T x (lR - {O})

which is bounded when

~(r,k)x

Let

T](r,k)

L

111l(r,k)11 ~ C(l + r)6

(12.16) with

for

(r

E

dr,k)

is a bounded

T)

k moves in a compact set in

is an X-valued, continuous function on

Tx (lR -~O}) x X. (i i)

We have

( 12. 1 7)

2i k( n(s , k) X,1 ( r, k) x' ) X ~ ({ G( r, S , k) - G( r, s , - k)} x ,x 1 ) X

for any

x,x'

C

X, k

E

lR - {OJ, and

(iii) The operator norm (12.18) with

C = C(k)

lR - {OJ.

I!:; (r,k)li n

Ilr1n(r,k)11 ~ C(l + r):)

(iv) Let

R moves in a compact set in

~ave

F](r,k)x,::: s - lim nn(r,k)x

(12.19) for any triple

is estimated as

(r e:: T,n=1,2, ... )

which is bounded when

Further, we

T

r,s,c

n

(r,k,x) f

C

E

in

X

)-lX)

I x (lR - {OJ) x X

L2,(~(I,X)

and let

k

C

lR - {O}.

Then

131

(1 2. 20)

in

PROOF.

(i) and (ii) can be easily obtained by proceeding as in the

proof of Theorems 10.4 and 10.8.

Since we have

q (r,k)x :: F (k)£.[s,kJ ,

(12.21)

n

n

Ilr~n(r,k)11

the uniform boundedness of nn(r,k)x

X.

to

n(r,k)x

and tile strong convergence of

follO\vs from Proposition 12.2.

Let us shO\v (iv).

It is sufficient to show (12.22)

~(r,k)f(r)dr

r

F(k)f ::

,/ I

fOr

f C L (I,X) 2 L ,we obtain

with compact support in

I.

Applying Theorem 10.5 to

n

/. (k) ::

(12.23)

n

(

J'

I

n ( r, k)f ( r) dr,

r1

whence (12.22) follows by letting

n ,.

ox:

•

Q. c. D.

Let

rl*(r,k)

with respect to

~l(r,k)

be tile adjoint of

11*(r,k)

obtained in

~10

Almost all tile results

are also valid for our

n*(r,k) . PROPOSITION 12.4. !l*(r,k) operator (i)

Let Assumption 12.1 be satisfied.

be as above and let nn(r,k) I.et

~,*(

'n

r, k)

associated with

RE:...IR-{O}.

Then

be the adjoint of the eigen-

Ln ~1*(·,k)xf~L2,_(Jl,X)

with the estimate (12.24)

Let

(x

i-:-

X) ,

forany

xcX

132

i'/here

C:: C(k)

is bounded \'ihen

k moves in a compact set in

IR-{O:.

He have

(12.25) (k

(i i )

Let

x

'1* ( r,

2) 1.26 (1 ~(r)

where

f_

lR - [0 ~, x ex, f (" L

k) x -2ikl._re - 1 .- ( ) i IJ ( r' , k) x-llr,k,x L ( • )

f (r)

o

(iii)

~

in

lR

(L - k2 ) ( . e i '

_L 2i k

C

-~O}

h(',k,x)

is the radiative

with

There exists

a compact set

( I, X)) .

lJ • Then

{L,-k,i'lfO]j

( 12• 27 )

~

2, "

is as in Proposition 10.7 and

function fOr

~

.! (

r' , k) x)

C(k) , which is bounded when

k

moves in

, sucn tnat (x c.: X) •

C 2. 28)

each

C

x c X (iv)

'l*(' ,k)x

o

2 (L - k )v(r)

(li.29)

n*(r,k)x

Hl ,8(I X)

'

='

0

loc

~'O(I)

(r c. I,k

and satisfies the equation C

is an x-valued, continuous function

PROOF.

IR-{O:,x 011

C

X) .

T x (lR·-{O}) x X

(i) and (ii) can be obtained by proceeding as in the proof

of Theorem 10.6 and Proposition 10.7, respectively. the proof of (i i i). from the relation

The uniform boundedness of

Let us turn into

:If! n (',k)xll -(,;:

follOlvs

133

(x,F (k)f)

n

(f

and the uni fOrm boundedness of

E

x

L2,~(I,X), n =

Now that (10.28) has been

II Fn (k)11

{n~(·,k)x}

established, in order to show the convergence of n*(·,k)x in

L _~(I,X)

2

,

1,2, ... )

it is sufficient to show for

to

xED

u

(12.31) in

L

Applying Proposition 10.7 to

,,(I,X)

2 , - ()

L , we have n

(12.32) fOr

x

function for {f 0 , n}

, where hn(·,k,x)

D , (r,k) <= T x IR-{O}

E

{Ln,-k"erfn]~

and

f

=

O,n

(L

can be eas i ly seen to converge to

n f

is the radiative

- k2 )(;e iu (r· ,k)x) 0

=

2 (L - k )( ~e i u ( r· , k) x)

L .. (I,X) , Theorem 4.5 is applied to show that {h (·,k,x)} 2 ,0 n to

h(·,k,x)

~~(r,k)x

L2,_cS(I,X), \'ihich implies (10.31).

in

satisfies the equation

(L

n

the proof of (iii).

it

converges

Noting that

2 - k )v = 0 , we make use of the

interior estimate and the convergence of in

show the convergence of

Since

{r*(· - In ' k)x} Hl ,8(I X)

o '

loc

in

L

2

,_6(I,X) to

which completes

We have

(12.33) (k

by letting

n -)-

cc

C:o:

IR-[O},

x

E

X, ,~Eo C(~(I,X))

in the relation

( 12. 34 ) ( n=1, 2, ••• , k

EO IR - {O}, x

ex, ¢

E

C'~ (I , X)) .

134

The first half of (iv) follows from (10.31) and Proposition 1.3. continuity of n*(r,k)x

Trle

can be shovm in quite the Sdllle \"Jay as in the

proof of Theorem 10.8, VJhich completes the proof.

Q. E. D. Now the Propositions 12.2, 12.3, 12.4 have been established, we can show the expansion theorem (Theorem 11.6) with the orthogonality of the general ized Fourier transforms

F±

(Theorerl 11.8) in quite the

same I'Jay as in the proof of Theorems 11.6 and 11.8.

Therefore "'Je shall

omit the proof of the following THEOREM 12.5.

Let Assumption 12.1 be satisfied.

alized Fourier transforms and the adjoint tively.

F~

F±*

t±

Then the gener-

is "'Jell-defined by Definition 11.4.

t±

can be represented as (11.16) and (11.17), respec-

are orthogonal and (11.21) holds good.

Concluding Remarks 1°

For the proof of the unique continuation theorem (Proposition

1.4) we referred to the unique continuation theorem for partial differential equation.

On the other hand

Jd~wr

131 gives the unique

continuation theorem for an abstract ordinary differential operator with operator-valued coefficients which can be referred to in our case. 2°

Our proof of the limiting absorption principle was alon~ In Ikebe-Saito III and Lavine III the

the line of Saito [31.

Schrodinger operator was directly treated and Ue 1 ililHin~ absorption principle for the Schrodinger operator with a long-range rotential was proved. 3° modifier

In §5 we introduced the kernel \(y.k)

D {2kZ(y) -

Noting that (13.2) 4°

Cb(y) - Z(y)2 - ;(yn ::: O(ly!-j-f') U:::O,l, t>1)

Z(y)2 + ~(y) . ;:), DJ f2k "ri-yT

(grad

- QO(Y)

~)2,

we can rewrite (13.1) as

2 ' .~ - (grad ),) } '"' O( !.YI- J -!)

(j--cO, L f'>l)

Our proof of Theorems 5.3 and 5.4 is a unification of the ones

in Saito [31 and 161. Jager [3J

of the stationary

as a solution of the equation

j

(13.1)

Z(y.k)

The method of th~ proof has its origin in

in which he treated an differential operator with orerator-

valued, short-range coefficients.

136

5°

In

~5

we defined the stationary modifier by a SOrt of

The condition (2 ) was assumed 0 so that the successive approximation process !!lay be effective.

successive approximation method.

As a result Our long-range potential 0-o(y) the estimates for the derivatives a rather large number.

is assumed to satisfy

DjQO~ j=O,1~2, ... ~m and m is

Here it should be mentioned that our proof of

Theorems 5.3 and 5.4 is effective as far as we can construct a stationary modifier VJhich satisfies (1) of Remark 5.9. [1 I constructed the time-dependent modifier

W(y,t)

Hormander

fOr a type of elliptic

operators with mOre general long-range coefficients than

ours.

Kitada

[2j showed that a stationary modifier satisfying (1) of Rerllark 5.9 can be constructed by starting with Hornlander's tinle-dependent modifier. (~)

If we use Kitada1s stationary modifier we can replace

by

~

(~)

There exist constants Co 4 is a C function and

~(y)

where

j D

and

0

< € ;

denote an arbitra~y derivative of

(j=1.2.3) • d(4)

>

1 such that

j-th

order and

d(j)

0 • d(l) + d(4) > 5 .

Gut in this lecture I adopted the primitive nlethod of successive approximation,

because we need further preparations with respect to the

theory of partial differential equation in order to introduce the more minute method which starts with Hormander Ill. 6° Kitada 7°

Theorem 8.7 was first stated and proved explicitly by [?] .

I kebe I 41 gave the proof of the orthogonality of the genera 1i zed

Fourier transforms by treating the Schrodinger operator directly and making use of the Lippmann-Schwinger equation.

Our proof of the

j+sO

137

orthogonality of F±

is different in usinq Proposition 8.5 instead

of the Lippmann-Schwinger equation. The modified wave operators.

8°

WD,~

wave operators

The time-dependent modified

for the Schrodinger operator with a long-range

potential we defined by Alsholm-Kato Ill, Alsholm III and GuslaevMatveev f 11 as (13.3) where

is a function of H On the other hand from the viewpoint O should of the stationary method the stationary wave operator vJD,~ Xt

be defi ned by ,v* _

(13.4)

-

F

WD,+_=

F,--'- F0 ,+

being the generalized Fourier transforms associated with

O,~

H ' O

From the orthogonality of the generalized Fourier transforms the completeness of WD,+ =

WD,~

WD,~

follows immediately.

is shown by Kitada fl 1, f21. [31

Recently the relation and Ikebe-Isozaki

fl], whence follows the completeness of the time-independent modified wave operator WD,+ go

In §12 we treated the case that

12l(y) by a sequence ~Qln(Y)L has compact support in RN Gut there is another

range potential. where

Qln(y)

Ql(y) is a general short-

Then we approximated

method which sta rts with the relations (13.5) where

(L_k 2 )-1 =(L _k 2)-1 n-C (L- k2 f l } l 1

2 L - - d 2 + B(r) + CO(r) l dr

,

is the identity operator and

138

v

=

(L_k 2 )-lf

r (k)

{L,k,~fl}. Then

means the radiative function for

can be defi ned by

Thi s method was adopted in I kebe I 31 . 100

In Theorem 11.9 we introduced the generalized Fourier

transforms

f~

associated with

H.

F

Let

be the generalized

O,~

Fourier transforms associated with of -/\.

H ' the self-adjoint realizatior O In this case the Green kernel GO(r~s~k) can be represented

by the use of the Hankel function and the exact form of

F

O~+

is

known as N

CI(N)(2~1)

(13.7)

=

(13.8)

C (N)

For proof see Saito r 21,

§ 7.

=

I

-~

l.i.m. R -)- 00

r I f IRe +"IY'¢(y)dy YI <

_e+i(N-l)i F

This means that

the usual Fourier transforms and the

-F

+

O~:!:

are essentially

are the generalization of

the usual Fourier transforms in this sense. lP

As was stated in the such as

Introduction~

lh-

an oscillating long-

range potential

%(y)

any assumption.

As for the Schrodinger operator with an oscillating

;(y) =

sin!yl

does not satisfy

long-range potential we can refer to Mochizuki-Uchiyama 11],12]. 12 0 Finally let us qive two remarks on the condition (5.2) in Assumption 5.1 on a long-range potential I? O.

In order to give

139

a unified treatment for in the case of ~

< E

~

0

< E ~

1 we assumed the condition (5.2).

we can adopt a weaker condition

But

mO = 2.

In fact in this case we have Z(y)

(13.9)

(y =

rw),

and the first condition of (5.41) can be weakened as (j=O,l,2),

(13.10)

because the condition (13.11)

is used to estimate the term

((Z' + P) u ,Bu)x

in the proof of

Lemma 6.5 only and we can directly estimate this term without making use of (1 3. 11 ) in the case of

12

<

c

~

1

Thus (5.2) can be replaced

by (13.12)

m > -2 -

o

E

(0 < c ;

~) and mo = 2

Next let us consider the case the i.e.

QO(y);- Q ( Iyl)· O

~(y)

In this case the stationary modifier

P(Y;A) are all identically zero.

Theorem 5.4 becomes much simpler. Lemma 6.6.

.

is sphericallY symmetric,

is also spherically symmetric and the operator M, 'P(y;A).

(~<E~l)

Z(y)

the functions

Therefore the proof of

For example, we do not need

Moreover (5.35), by which

Z(y);s defined, takes the

following simpler form:

(13.13)

and we set

Z(y) ;- Z where nO

nO

is the least integer such that

140

(no + l)c > 1.

Noting that the right-hand side of (13.13) does not

contain any derivative of

Zn(y) ,

(13.14)

(if QO(Y) is spherically symmetric).

m = 2

o

we can replace (5.2) by

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Spectral properties of Schrodinger operators and scattering theory, Ann. Scuola NOr. Sup. Pisa (4) 2 (1975), 151-218.

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Wave operators fOr long-range scattering, Thesis, UC Berkeley ( 1972) .

P.K. Alsholm and T. Kato [1]

Scattering with long-range potentials, Proc. Symp. Pure Math. Vol XIII (1973), 393-399.

N. Aronszajn [1]

A unique continuation theorem for solutions of elliptic partial differential equations Or inequalities of second order, J. de Math. 36 (1957), 235-247.

E. Buslaev and V. B. Matveev [1]

Wave operators fOr the Schrodinger equations, with a slowly decreasing potential, Theoret. and Math. Phys. 2(1970), 266-274.

J . D• Do 11a rd

[1]

Asymptotic convergence and the Couloillb interaction, J. Mathematical Phys. 5 (1964), 729-738.

A. Erdelyi and others [1]

Higher transcendental function II, McGraw-Hill, New York, 1953.

L. H1:lrmander [1]

The existence of wave operators in scattering theory;

Math.

Z. 149(1976), 69-91.

T. I kebe [1]

Eigenfunction expansions associated with the Schrodinger operators and their application to scattering theory, Arch. Rational Mech. Anal. 5(1960), 1-34.

[2J

Spectral representations fOr the Schrodinger operators with long-range potentials, J. Functional Analysis 20 (1975), 158-177 .

[3]

Spectral representations for the Schrodinger operators with long-range potentials, II, Publ. Res. Inst. Math. Sci. Kyoto Univ. 11 (1976), 551-558.

142

[4]

Remarks on the orthogonality of eigenfunctions for the Schrodinger operator in Rn J. Fac. Sci. Univ. Tokyo 17 (1970),355-361.

1. Ikebe and H. Isozaki

D] Completeness of mOdified wave operators for long-range potentials, Preprint.

T. Ikebe and T. Kato [1]

Uniqueness of the self-adjoint extension of singular elliptic differential operators, Arch. Ratinal Mech. Anal. 9 (1962), 77-92.

T. Ikebe and Y. Saito [1]

Limiting absorption method and absolute continuity for the Schrodinger operator, J. Math. Kyoto Univ. 7(1972), 513-542.

W. Jager [1]

Uber das Dirichletsche Aussenraumproblem fUr die Schwingungsgleichung, Math. Z. 95 (1967), 299-323.

[2]

Zur Theorie der Schwingungsgleichung mit variablen Koeffizienten in Aussengebieten, Math. Z. 102 (1967). 62-88.

[3]

Das asymptotische Verhalten von Losungen eines typus von Differential gleichungen, Math. Z. 112 (1969), 26-36.

[4]

Ein gewohnlicher Differentialoperator zweiter Ordnung fUr Funktionen mit Werten in einem Hilbertraum, Math. Z. 113 (1970) 68-98.

1. Ka to

[1]

Perturbation Theory fOr Linear Operators, Second Edition, Springer, Berlin, 1976.

H. Kitada [1]

On the completeness of modified wave operators, Proc. Japan. Acad. 52 (1976), 409-412.

[2]

Scattering theory fOr Schrodinger operators with long-range potentials, I, J. Math. Soc. Japan 29 (1977), 665-691.

[3]

Scattering theory fOr Schrodinger operators with long-range potentials, II, J. Math. Soc. Japan 30 (1978), 603-632.

143

S. T. Kuroda [1]

Construction of eigenfunction expansions by the perturbation method and its application to n-dimensional Schrodinger operators, MRC Technical Summary Report No. 744 (1967).

R. Lavi ne [1] Absolute continuity of positive spectrum for Schrodinger operators with long-range potentials, J. Functional Analysis 12 (1973), 30-54. K. Mochizuki and J. Uchiyama

Y. Sa

[1]

Radiation conditions and spectral theory fOr 2-body Schrodinger operators with "oscillatory" long-range potentials, I, the principle of limiting absorption, to appear in J. Math. Kyoto Univ.

[2]

Radiation conditions and spectral theory fOr 2-body Schrodinger operators with "oscillatory" long-range potentials, II, spectral representation, to appear in J. Math. Kyoto Univ.

ito [1] The principle of limiting absorption for second-order differentia equations with operator-valued coefficients, Publ. Res. Inst. Math. Sci. Kyoto Univ. 7 (1971/72), 581-619. [2]

Spectral and scattering theory fOr second-order differential operators with operator-valued coefficients, Osaka J. Math. 9(1972), 463-498.

[3]

Spectral theory for second-order differential operators with long-range operator-valued coefficients, I, Japan J. Ma th. New Ser. 1 (1975), 311- 349.

[4]

Spectral theory for second-order differential operators with long-range operator-valued coefficients, II, Japan J. Math. New Ser. 1 (1975), 351-382.

[5]

Eigenfunction expansions for differential operators with operator-valued coefficients and their applications to the Schrodinger operators with long-range potentials, International Symposium on Mathematical Problems in Theoretical Physics. Proceedings 1975, Lecture Notes in Physics 39(1975), 476-482, Springer.

[6]

On the asymptotic behavior of the solutions of the Schrodinger equation (-6 + Q(Y)-k 2 )V ::: F, Osaka J. Math. 14(1977), 11-35.

1~4

UJ

Eigenfunction expansions for the Schr~dinger operators with long-range potentials Q(y) = o( IYI-C), E > 0, Osaka J. Math. (1977), 37-53.

D. W. Thoe [lJ

Eigenfunction expansions associated with Schr~dinger operators in Rn, n > 4, Arch. Rational Mech. Anal. 26(1967), 335-356.

K. Yosida [lJ

Functional Analysis, Springer, 1964.

NOTATION INDEX

A, AN

12

B( r)

6

B( Y, z)

12

B(X)

12

b , b j (8) j

56

(C+

10

C(r)

6, 21

Co (r) , C1 (r)

21

C(Cl,

~,

••

00

0

)

12

Co (J, X)

10

Cae ((T, X)

11

D

12

rfi

I?

D( J)

18

V(w)

12

E(0

F+,

,

M)

F;

117 119

146

125 f(k)

87 119 125

54 F ~ ( J, X), F ( J, X)

11

G(r, s, k)

101

H

116

H6:~(J, X), H6,B(J, X)

10

H1 ,B(I

12

o

'

X)

10c

12

I, I

10

L

6 10 12

147

M

57, 117

mO' m1

53

M

j

56

P(y)

57

T

1

U

5

v( • , k, Q)

23

,.., WD, +' WD,+

137

X

5, 10

Y(y,

k)

62 55, 62

Z (y, k) ~

ak(r), ak(r)

80

"Q(r, k), 7* (r, k)

106

,?±(r, k) ,

-

Q±(r, k) ,

17; (r,

k)

119

~:(r,

k)

125

148

AN

12

'A(y, k)

55, 62

~

(y, k)

54

Cf(y; 'A)

57

SUBJECT INDEX

Eigenoperator

106

Generalized Fourier transform

4, 119

Green formula

96

Green kernel

101

Interior estimate

16

Laplace-Beltrami operator

6, 56

Limiting absorption

8, 22

princi~le

Long-range potential

4

Orthop-;oDa.lity

123

Polar coordinates

6, 56

Radiative function

22

Regularity theorem

17

Schrodinger operator

1

Short-range potential

4

Spectral representation theorem

121

Stationary modifier

9, 56

Stationary wave operator

137

Time-dependent modifier

136

Time-dependent modified wave operator

137

Unique continuation theorem

19

Spectral Representations for Schrödinger Operators with Long-Range Potentials (Lecture Notes in Mathematics)