Singular Sturm-Liouville problems: The Friedrichs extension and comparison of eigenvalues

SINGULAR STURM-LIOUVILLE PROBLEMS: THE FRIEDRICHS EXTENSION AND COMPARISON OF EIGENVALUES H.-D. NIESSEN and A. ZETTL [Received 6 August 1990—Revised 21 March 1991]

ABSTRACT

A new characterization of singular self-adjoint boundary conditions for Sturm-Liouville problems is given. These are an exact parallel of the regular case. They are given explicitly in terms of principal and non-principal solutions. The special nature of the Friedrichs extension is clearly apparent and highlighted. Inequalities among the eigenvalues of different boundary conditions, separated and coupled, are obtained. Most of all we want to stress the method of proof. It is based on a very elementary transformation which transforms any singular non-oscillatory limit-circle endpoint into a regular one.

1. Introduction The purpose of this paper is to show that to every Sturm-Liouville problem which is bounded below there exists a very simple transformation, namely multiplication by a function, which transforms it into another Sturm-Liouville problem in such a way that every (finite) limit-circle endpoint is transformed into a regular one. Then many basic properties of singular problems can be inferred from corresponding regular ones. For example, we show that the characterizations of the Friedrichs extension of singular Sturm-Liouville problems in terms of principal and also in terms of non-principal solutions, due to Rellich [13] and Kalf [8] follow easily from the regular case. The inequalities among the eigenvalues for different boundary conditions given by Jorgens in [7] and the recent ones given by Weidmann in [15], both for regular problems, are extended to singular non-oscillatory eigenvalue problems in the limit-circle case. These seem to be new. Also, the asymptotic formula for the distribution of the eigenvalues which is well known for regular problems (see, for example, [1]) is generalized to these singular eigenvalue problems. All this follows readily from our transformation because with this transformation it is apparent which singular problem transforms into which regular one. The organization of the paper is as follows. After the introduction in § 1, principal and non-principal solutions are discussed in § 2, followed in § 3 by the study of the transformation Tv which serves as a main tool. In § 4 the relationship between semi-boundedness of the minimal operator, oscillation, and the Friedrichs extension is studied. The comparison of eigenvalues and the asymptotic formula for the distribution of the eigenvalues is given in §5, examples are discussed in § 6, and § 7 is devoted to the proof of Lemma 2.3. This proof is given here because it is long and technical. Part of this work was done while the second-named author was a visiting Professor at the University of Essen. His stay was supported by a DFG-grant. 1991 Mathematics Subject Classification: 34B24, 34C10, 47A20, 47E05. Proc. London Math. Soc. (3) 64 (1992) 545-578.

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H.-D. NIESSEN AND A. ZETTL

2. Principal and non-principal solutions Let M be the differential expression defined by Afy : = - [ - ( / ? / ) ' + qy] onl = (a,b),

with - » *£ a < b ^ ».

(2.1)

Throughout this paper we assume that p,q,w:

/-»R,

p,tv>0a.e,

- , q, w e L 1OC (/). P

(2.2)

Consider the equation A/y = A_y, that is, -(py'Y

+ = ^wy

on I, with A e R.

(2.3)

The local integrability conditions of (2.2) ensure that the zeros of any non-trivial solution of (2.3) inside the interval / are isolated; thus they can accumulate only at an endpoint. If the zeros of one (and hence of every) non-trivial real-valued solution of (2.3) accumulate at the endpoint a (or b), then (2.3) is called oscillatory at a (respectively b), otherwise (2.3) is called non-oscillatory at a (respectively b). DEFINITION 2 . 1 . (See [6].) Let AeIR and let u, v be real solutions of (2.3). Then (a) u is called a principal solution at a if

(i) u(t) =£ 0 for t in {a, a) and some a in /, (ii) every solution y of (2.3) which is not a multiple of u satisfies u(t) = o(y(t)) asf->a.

(2.4)

(b) v is called a non-principal solution at a if (i) v{t) ¥=0 for t in (a, a) and some a in /, (ii) v is not a principal solution at a. Principal and non-principal solutions at b are defined similarly. To simplify the notation we state definitions and assertions only for the left endpoint a. Similar definitions and assertions always hold, and are freely used, for the right endpoint b. The notion of 'principal solution' has been introduced in [10]. The properties of principal and non-principal solutions have been investigated by Rellich [13]. Immediately clear from the definition is: 2.1. Let keU. If (2.3) has a principal solution u at a, then every non-zero real multiple of u is also a principal solution at a and no other solution is a principal solution at a. LEMMA

REMARK 2.1. By Lemma 2.1 the principal solution at an endpoint, if it exists, is unique up to a real constant multiplicative factor. Simple examples show that the same solution may be principal at one end and non-principal at the other end. Non-principal solutions are never unique, since if v is a non-principal solution and u is a principal solution (which in this case exists by Theorem 2.1) then v + cu is also a non-principal solution for any c in U. Clearly principal and non-principal solutions do not exist at an endpoint at which the equation (2.3) is oscillatory.

SINGULAR STURM-LIOUVILLE PROBLEMS

547

REMARK 2.2. If M is regular at a (see § 3 for the definition of regular) then for any solution y of (2.3), y and py' can be continuously extended to a and principal solutions u at a exist and satisfy the initial conditions: u(a) = 0, (/?«')(<*)=£ 0. Similarly any non-principal solution v at a satisfies v{a) =£ 0. LEMMA 2.2. Let XeU, J be a subinterval of I, and y, z be linearly independent solutions of (2.3) on J with y(t)^0 for teJ. Then c =y(pz') — z(py') is a non-zero constant which is real if y and z are real and

<•>©'«•:Pit)y\t)

, forteJa.e.,

(2.5)

(ii) ify and z are real, then z/y is monotone on J, -£2' py2

wherea.teJ.

(26) (2.1)

Proof. Clearly (ii) and (iii) follow from (i), and (i) is an immediate consequence of (2.3) and the definition of c. THEOREM 2.1. Let A e U. Then (2.3) is non-oscillatory at a if and only if there exists a principal solution at a.

Theorem 2.1 and the following Theorem 2.2 (apart from (2.11)) can be found in [13] (with more restrictive assumptions on the coefficients of M). Nevertheless, since our proof is much simpler, we will give it here. Proof. Since by definition a principal solution at a is non-oscillatory at a, we need only show the converse. Let yx and y2 be linearly independent real solutions of (2.3). Since (2.3) is non-oscillatory, it follows from the Sturm separation theorem that there is an oc in / such that y\{t), y2(t) =£ 0 for a < t < a. From (2.6) it follows that lim ^ 7 7 = L, with - « ^ L t-*a

y2(t)

exists. If L is finite, define u\=yx — Ly2, v :=y2; otherwise, define u :=y2, v := yt. In any case, u and v are real linearly independent solutions of (2.3) such that

Let y be any solution of (2.3) which is not a multiple of u. Then there exist constants c, d such that d =£ 0 and y = cu + dv. Therefore, lim

u(t)

= lim c + d

v(t) u(t)

Hence u(t) = o(y(t)) as t—>a. This shows that u is a principal solution at a.

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H.-D. NIESSEN AND A. ZETTL

THEOREM 2.2. Assume that (2.3) is non-oscillatory at a for some A in U. Let u, v be real non-trivial solutions of (2.3) and let a el be such that u{t), v(t) =£ 0 for a
r i

— z = 00; Ja pu pu

(2.8)

(ii) v is a non-principal solution at a if and only if (29

> pv (iii) if u is a principal solution and v a non-principal solution at a, then there exists c inU, with c =£ 0, such that

('

u(t) = v(t)

C

—2>

where

a
(2.10)

and \u(t)v(x)\<\u(x)v(t)\, wherea
(2.11)

Proof (i) Let y be a solution of (2.3) which is linearly independent of u. Then by (2.7), c

u(t)

u(a)

2

{ a < t

^

a )

2

J, pu

Therefore, the divergence of j£ l/pu2 is equivalent to the condition

that is, to u(t) = o(y(t)) (f-»a). By Definition 2.1, this is equivalent to u being a principal solution. (ii) Since every real non-trivial solution of (2.3) is either a principal or a non-principal solution, (ii) is merely a restatement of (i). (iii) By (2.7), there exists a real non-zero constant c such that u(t) u{oc) r' cc _u(t) 2 _ Lpv v(t) pv2

v(t) v(a)' Letting a —» a we get (2.10). Using (2.10) twice, we obtain, for a < t <x ^ a,

\u(tMx)\ = \v(t)v(x)c\ [J I <| u (r) u (x)c| \—2 = \v(t)u(x)\. Ja P*)

Ja P*J

The operators associated with the differential expression M are studied in the Hilbert space L2w(l) with inner product (y, z)w := yzw. h

SINGULAR STURM-LIOUVILLE PROBLEMS

549

For any subinterval J of I let ACloc(7) denote the set of complex-valued functions on / which are absolutely continuous on all compact subintervals of /. Define the 'global domain' GP(J) of M (with respect to / ) by GP(J) := {y e AC loc (/)| py' e AC loc (/)}. Then My is defined and is in LwAoc(J) for all y in GP(J). Define D(M, J) := {y e GP(J)\ v, My e L2W(J)}. Then it is known [11] that D(M, J) is dense in L2W{J). We will denote D(M, I) by D, or by D(M) when we want to emphasize the dependence on M. Define Mmaxy := My for y e D and let M>:=M*ax and DQ:= D(M0). The operators Mo and A/max are called the minimal and maximal operators of M in Li,(I). It is known [11] that Mo is the closure of the symmetric operator MQ defined by restricting M to the domain Do:= D(MQ) := {y e D\ y has compact support in / } . The operator MQ is called the pre-minimal operator and DQ the pre-minimal domain of M in L2W(J). The following lemma is needed several times in the present paper. It sharpens the well-known result that, for real A, B, A', B' and a 0 there exists a real-valued function y e D(M, [a, /S]) such that (a)y(a) = A, y(p) = B, (py')(a) = A', (py')(fl) = B', (b) mm(A, B) - e ^y(x) ^ max(>4, B) + e for x e [a, 0],

( c)/V.| B -^(/T + , Since the proof of this lemma is rather involved and technical, it is postponed to §7. 2.3. (See Rellich [13], Rosenberger [14].) Suppose MQ is bounded below with lower bound a. Then for any A < oc every non-trivial real solution y of (2.3) has at most one zero in (a, b). THEOREM

Rellich states this theorem under the additional assumptions that p is continuously differentiable and that q, w are piecewise continuous. (As he gives it, his proof is correct only for a 5=0.) Rosenberger treats the general case considered here. He uses a detailed investigation of the energy space of MQ to proof the theorem. Here, for the general case, we give an elementary proof in the spirit of Rellich.

550

H.-D. NIESSEN AND A. ZETTL

Proof. Suppose there is a non-trivial real solution y of (2.3) with A < a satisfying

y(s) = 0 = y(r), for a

<s
Without loss of generality, we may assume that a = 0 since, otherwise, we can replace q with q — aw. Then we choose o e (a, s), A := B := A' := 0, B' := (py)'(s) and e > 0 so that

Applying Lemma 2.3 to [o, s] and to these values of A, B, A', B' and e, we obtain a real-valued function u such that U €

D{M, [°> s])>

u(o) = u(s) = (pu

(pu')(s) = (py

[pu'2^

\u(x•)\«* (x e [o, *]),

e.

This implies that

f {pu"

' + \q\u2}^E

+ E2

F

F

2

w.

J rw

Similarly, we choose p e (r, ^), >1 := 5 := fi' := 0, A' := (p.y)'(r) and e > 0 with e + e2j^\q\<-ikirsy2w, and apply Lemma 2.3 to [r, p], A, B, A',B', e to obtain a real-valued function v such that

veD(M,[r,p]), v(r) = v(p) = (pv')(p) = 0, (pv')(r) =

Now let

'0 for a < t < o, u(t) for o z{t) = y(t) for s
for p < t < b.

Then z belongs to the pre-minimal domain DQ of M and

(Afz, z)^ = [ z(Mz)w + [ z(Mz)w + A [ /W Ja

Js

Jr

= f {pu'2 + qu2} + \P {pvt2 + qv2} + k \ y2w<0, J<j

contrary to the assumption (Mz,

*r

z)w^0.

Js

SINGULAR STURM-LIOUVILLE PROBLEMS

551

Theorem 2.3 and Theorem 2.1 immediately imply COROLLARY 2.1. Suppose MQ is bounded below with lower bound a. Then for any A < oc, (2.3) is non-oscillatory at a and b, and hence has principal solutions at a and at b.

3. The transformation Tv Let v be a positive function on /. Then for every complex-valued function y on / we define Tvy-ylv. 3.1. Let v be a positive function belonging to G p (/) and let P := v2p, W := v w. Then GP(I) = GP(I) and Tv maps (1) GP{1) bijectively onto GP(I), (2) £w,ioc(/) bijectively onto LWAoc(I), (3) L2W{I) isometrically onto L^(/). The inverse maps are restrictions c/ (7J,)"1 = 7\,-i. LEMMA 2

Proof Obviously, Tv is injective and (7^) -1 = 7^-i. Furthermore, it is easy to see that Gp(I)czGP(I) and that (l)-(3) hold with 'bijectively' replaced by 'injectively' and with 'onto' replaced by 'into'. But since v~l is positive and belongs to GP(J), these results applied to P and Tv-\ instead of p and Tv prove the lemma. 3.2. Let v be a positive function belonging to GP(I) and, with T :=TV, define N := TMT~X, P := v2p, W := v2w, Q := vwMv. Then N has global domain GP{I) and (1) Nz = W-l[-(Pz')' + Qz] for z e GP(I), (2) P, W>0a.e., P~\ Q, WeL loc (/), (3) W,Q<=L(I)ifv, MveL2w(I), (4) Nmax=TMmaxT-\ (5) N(>=TM^T-\ (6) N0=TM0T-\ (7) (NmaxTylt Ty2)w = (Mmaxylt y2)w for yuy2e D(M), (8) [Tyu Ty2]N = [yi, y2]M for yuy2e D(M). Here [yl} y2]M =y\{py2) — (pyiWi denotes the Lagrange form of M and similarly, [z\> Z?\N = Zi(Pz2) ~ iPz'x)^ denotes the Lagrange form of N. LEMMA

Proof (1) For zeGP(I), T~xz = vz belongs to GP(I) by Lemma 3.1(1). l Therefore, Nz = TMT~ z is defined and Nz = (uw)- 1 [-(p(uz)')' + qvz] = (mv)"^- (pu 2 u~V +pzv')' + qvz] = (vw)~x[- (puV)'ir 1 +pv'z' -pz'v' - (pv')'z + qvz] = W" x [- (Pz1)' + v(- (pv'Y + qv)z]

552

H.-D. NIESSEN AND A. ZETTL

Furthermore, by this equation and the above definition, GP{I) is the global domain of N. (2) Since p~l and w are positive a.e. and locally integrable, the same holds true for P~l and W. Furthermore, Mv belongs to Lwioc(I) for v e GP(I). This implies that Q is locally integrable, too. (3) This follows immediately from the definition of W and Q. (4), (5), (7) By definition of D(N), D(M) and by Lemma 3.1(1), (3) we get z,NzeL2w(I)}

D(N) = {zeGP(I)\

= T{y e GP(1)\ Ty, TMy e L2W(I)}

(3.1)

2

= T{y e GP(I)\ y, My e L W{1)} = TD(M). Since D(N) and D(M) are the domain of definition of Nmax and A/max, respectively, (4) follows from yV= TMT~l. By (3.1), also D(NQ) = {ze D(N)\ supp(z) is compact in (a, b)} = T{y e D(M)\ supp(_y) is compact in (a, b)} =

TD(MQ).

This in turn implies (5). Furthermore, for yx, y2e D(M) we get, by (4) and Lemma 3.1(3), (NmaxTylt Ty2)w = (TMmaxyx, Ty2)w = (Mmaxyx, y2)w. (8) Since v is real, for yx, y2 e D(M) we get [Tyu

~~ v -7x

PV>

y) -—yM

J —

L /

v ,x—

PV>

—

- (py^2 + —yw

(6) Since Mo and A^o are the adjoints of Mmax and Nmax, respectively, we obtain by (3.1) and (7), D(N0) = {ze D(N)\ (Nz, z,)w = (z,Nzx)w for all zx e D(N)} = T{y 6 D(M)\ (My, yx)w = (y, Myx)w for all yx e D(M)} = TD(M0). Therefore, (6) follows from (4). DEFINITION 3.1. Suppose S is a densely defined symmetric (but not necessarily closed) operator in a Hilbert space (//, ( , )) which is bounded below. Let D(SF) be the set of all y e D(S*) for which there exists a sequence yn in D(S) such that

(1) yn -» y in H as n -» °°, (2) (S(yn -ym), yn - ym)-*0 Define the operator SF by

as n, m-»•».

SFy = S*y,

for y e D(SF).

SINGULAR STURM-LIOUVILLE PROBLEMS

553

Then SF is called the Friedrichs extension of 5. According to the well-known result of Friedrichs [4], SF is a self-adjoint operator with the same lower bound as S. COROLLARY 3.1. Let v, T :=TV and N be as in Lemma 3.2. Furthermore, let Mx be a symmetric extension of MQ and A^ := TA^r" 1 . Then Nx is a symmetric extension of NQ and Nx is bounded below if and only if Mx is bounded below. In this case we have for the Friedrichs extensions NF and MF of Nx and Mx,

respectively,

Proof The fact that NX = TMXT~1 and Lemma 3.2(7) show that Nx is a symmetric extension of A^ and that Nx is bounded below if and only if M, is bounded below (with the same lower bound). Furthermore, as in the proof of Lemma 3.2(6), we get D(NX*)=TD(MX). Therefore, using Lemma 3.1(3) and Part (7) of Lemma 3.2, we obtain D(NF) = {ze D(Nf)\ 3zn e D(NX) such that zn^>z in and (N(zn - zm), zn - zm)w-•

0 (n, m -> <»)}

= {7>| ye D(M?), 3yn e D(MX) such that Tyn^>Ty in L2W(I) and (NT(yn - ym), T(yn - ym))w -* 0 (n, m -+ «>)} = {7>| y e D{Mt), 3yn e D(MX) such that yn^y and (M(yn -ym), (yn -ym))w^>0

in L2W(I)

(n, m-»oo)}

= TD{MF). Therefore, the assertion follows from Lemma 3.2(4). In particular, in view of Lemma 3.2(5), Corollary 3.1 is applicable to Mx = MQ, NX = NO. The expression M is said to be in the limit-circle case at a if all solutions of (2.3) are in L2w(a, c) for some (and hence any) c in (a, b). It is known [11, § 19.4, Theorem 4] that this classification is independent of A. Otherwise M is said to be in the limit-point case at a. We call M pseudo-regular at a if p~x, q, w are in L(a, c) for some (and hence any) c in (a, b); M is said to be regular at a if a is finite and M is pseudo-regular at a. REMARK 3.1. Simple calculations show that every locally absolutely continuous and strictly increasing transformation x =f{i) of the independent variable preserves the properties of being pseudo-regular, bounded below, oscillatory, limit-circle and being a principal or non-principal solution at some endpoint considered. If this transformation is chosen to admit a bounded inverse (for example, x = tan t) then, in addition, the endpoints of the transformed equation become finite. This shows that a pseudo-regular endpoint can always be transformed to a regular endpoint without affecting the remaining abovementioned properties. (Compare also [3].)

554

H.-D. NIESSEN AND A. ZETTL

Using such a transformation, the following remark is implied by § 16.2 and § 17.3 of [11] (compare Remark 2.2). It can also be proved directly by using Gronwall's inequality and the variation-of-constants formula, both applied to the first-order differential system which is equivalent to (2.3). REMARK

3.2. If M is pseudo-regular at a, then lim_y(;t) and lim (py')(x) x—*a

x—*a

exist and are finite for any solution y of My = Ay, A in C and also for any y in D{M). Furthermore, these limits can assume any pre-assigned complex values for a suitable solution y of My = ky. This implies, in particular, that M is in the limit-circle case at a and that My = ky is non-oscillatory at a for each real A. 3.3. Let v be a non-principal solution for (2.3) at a {which implies (2.3) to be non-oscillatory at a). Then M is in the limit-circle case at a if and only if v belongs to L2w{a, c) for some c in (a, b). LEMMA

Proof To prove the lemma let u be a principal solution of (2.3) at a. Such a principal solution exists by Theorem 2.1. Since u and v are linearly independent and u(t) = o(v(t)) for t^>a, all solutions of (2.3) belong to L2w(a, c) (for some c in {a, b)) if and only if v belongs to L2w{a, c). 3.2 (Rellich [13]). / / (2.3) is non-oscillatory at a for some real A, and if V(w/p) is not integrable at a, then M is in the limit-point case at a. COROLLARY

Proof By Theorem 2.1, there exists a non-principal solution v of (2.3). If M is in the limit-circle case, then p

V(pu 2 )

is integrable at a by Theorem 2.2(ii) and Lemma 3.3. 3.4. Let v, T and N be as in Lemma 3.2, and let A be real. Then (1) M is in the limit-circle case at a if and only ifNis in the limit-circle case at a, (2) My = ky is non-oscillatory at a if and only if Nz = kz is non-oscillatory at a.

LEMMA

Proof. Since y is a solution of My = ky if and only if Ty is a solution of Nz = kz, (2) is obvious and (1) follows from Lemma 3.1(3) (with b replaced by ce(a,b)). LEMMA 3.5. Let v e GP{I) be a positive function which for some a el and for some real k is a solution of My = ky on (a, a]; let T, P, W, Q, N be as in Lemma 3.2. Then (1) QW~l is constant on (a, a]. If the restriction of v to {a, a] is a non-principal solution of My = ky at a, then (2) p-leL{a,c)force{a,b), (3) N is in the limit-circle case at a if and only if N is pseudo-regular at a.

SINGULAR STURM-LIOUVILLE PROBLEMS

555

Proof. (1) By Lemma 3.2, Q = vwMv = kv2w = kW on (a, <x). (2) P" 1 = {v2p)~l is integrable on (a, a] by Part (ii) of Theorem 2.2. Being locally integrable by Lemma 3.2(2), P" 1 belongs to L(a, c). (3) By Lemma 3.4(1) and Lemma 3.3, N is in the limit-circle case if and only if v belongs to L2w{a, c), that is, if and only if W belongs to L(a, c). By (1), (2) and Lemma 3.2(2), this is equivalent to P~\ Q, W e L(a, c). LEMMA 3.6. Let a
Proof. Applying Lemma 2.3 with e < min(ya(a), yb(f5)), we obtain a positive function y e Gp([a, /?]) such that

y(a)=ya(a), y(p)=yb(P), («0'(«) = (P>0(«), Now let ya(x)

for a < x ^ a,

!/(*) = y(x)

fora<x
for Then v is in AC|OC(/) for / = (a, a), J = (a, fi), J = (j8, b) and v is continuous at a and at p. Hence v is in ACloc(/). Similarly, we get that pv' e AC|OC(/). Thus veGp(I). COROLLARY 3.3. //, for some real A, (2.3) is non-oscillatory at the endpoint a, then there exists a positive function v e GP(I) which coincides near a with a non-principal solution va of My = Ay at a. If, in addition, M is in the limit-circle case at a, the transformed operator N := TvMTZl is pseudo-regular at a, and hence it is regular at a if a is finite.

Proof. By Theorem 2.1, there exists a non-principal solution va of My = Ay at a, which may be assumed to be positive on some interval (a, a], with a
ofN(>. tfya and yb are non-principal solutions at a and b, respectively, then (3) for zeD(N), (\/P)z' e L2(I) and lim^^^z^), \imx^bz(x) exist and are finite, (4) the domain of the Friedrichs extension NF is determined by D(NF) = {ze D(N)\ lim z(x) = lim z(x) = o). I

x->a

x-*b

J

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H.-D. NIESSEN AND A. ZETTL

Proof. By Lemma 3.5(1) there exist a, {5 e(a,b) with oc
r

W\z\2.

Then, for z e D{NQ), we obtain

(P|zf P\z '\2 W\z\2+\

-K2\ Ja

{P\z'\2-KxW\z\2).

Jp

With K := max(^!, K2) this implies that P\Z'\2-K\

{Nfr,z)w&\\ •^ Ja

W\Z\2

forzeDW).

(3.2)

Ja

Therefore, NQ is bounded below (with lower bound — K). If z belongs to D(NF), then by Definition 3.1 there exists a sequence zneD(Nfi such that z w -»z in L 2 ^/) and (N^zn - zm), zn -zm)w-+0 (#i->«>). Therefore, by (3.2), (VP)z^ is a Cauchy sequence in L2(I) converging to some h e L2(I). For / := [c,rf]c /, 1/VP e L 2 (/) by Lemma 3.2(2). Hence, u := belongs to L(J) and z'n=^{\/P)z'n^u

in

Since zrt converges to z in L\{1), we may assume without loss of generality that zn converges to z a.e. on /. Let f eJ be such that z n (|)—>z(|) for n—»«>. Then 2(x) - z ( | ) = Hm [zM(x) - z n (|)] = lim f z,;(/) dt = \ u{t) dt,

(3.3)

first a.e. on /, then, since both sides of (3.3) are continuous, everywhere on /. Therefore, z' = u a.e. on J. Since J is an arbitrary compact subinterval of /, this holds on / as well. Hence, {y/P)z' = (VP)w = h e L 2 (/). Now let ya and yfc be non-principal solutions at a and b, respectively. Then, by Lemma 3.5(2), 1/VP € L\l) and the considerations leading to (3.3) now imply that

and, since zn vanishes in some neighbourhood of a, z(x) = lim zn{x) = lim I z'n(t) dt = I u{t) dt. n—»oo

n—+°° Ja

Ja

SINGULAR STURM-LIOUVILLE PROBLEMS

557

Thus, \imx^az(x) = 0 for z e D(NF). Similarly, \imx_>bz(x) = 0 can be proved. Therefore, in this case, limz(*) = limz(jc) = 0 for z e D(NF). x—*a

(3.4)

x—*b

Now let z belong to D(N). Then Lemma 2.3 implies that there exists za e D(N) which coincides with z in (a, a] and which vanishes in [/J, b). Then zb\—z — za belongs to D(N) as well, vanishes in (a, a], and coincides with z in [/?, b). If N is in the limit-circle case at a, then, by Lemma 3.5(3), N is pseudo-regular at a. Therefore, Remark 3.2 shows that linr^,, za(x) and lim^._»fl (Pza)(x) exist and are finite. Furthermore, {y/P)z'a = {\NP)Pz'a belongs to L2(/)> since l/VPeL 2 (/) and za vanishes in [/?, b). If Af is in the limit-point case at a, then za belongs to D(N0) and therefore to D(NF), too. Hence, by (2) and (3.4), (VP)z« belongs to L\l) and limx^flzfl(*) exists and is finite. Since za vanishes in [jS, b), linv_ 6 za(x) obviously exists. This shows that the assertion (3) holds for za. Similarly, it can be proved that zb, and therefore z = za + zb, fulfills (3). To prove (4), let

Dx:=\zeD(N)\ I

lim z{x) = lim z(x) = o|. x-*a

x—*b

)

If N is in the limit-circle case at a then from Lemma 3.5(3), Remark 3.2 and the definition of D1} it follows that [zx, z2]N(a) = 0 for all zlf z2e Dx. UN is in the limit-point case at a, then [zx, z2]N(a) = 0 for all zx, z2eD(N) [11, p. 78]. Similarly, [zx, z2]N(b) = 0 for zx, z2eDx. Therefore, the restriction S of N to Dx is symmetric. By (3.4), the Friedrichs extension NF of NQ is a restriction of 5. Hence, D(NF) = DX. 4. Oscillation, semiboundedness and the Friedrichs extension We start with a couple of abstract lemmata. LEMMA 4.1. Suppose S and T are symmetric densely defined operators in a Hilbert space such that SczT and dimD(S*)/£>(5)<°°, where S denotes the closure of S. Then S is bounded below if and only if T is bounded below.

Proof. If T is bounded below then obviously S is bounded below. If 5 is bounded below, let R denote a maximal symmetric extension of T. Then R is a maximal symmetric extension of 5. Since S has a self-adjoint extension (namely the Friedrichs extension), its deficiency indices must be equal and therefore R is self-adjoint. Furthermore, 5 is bounded below and R is a self-adjoint extension of 5. Therefore, by § 14.11 of Naimark [11] and the assumption dim D(S*)/D(S) <<»,/? is bounded below. This in turn implies that T is bounded below. This lemma in particular shows that every symmetric extension of MQ is bounded below if and only if MQ is bounded below. The following lemma then guarantees, for example, that the Friedrichs extensions of A/Q and Mo are the same.

558

H.-D. NIESSEN AND A. ZETTL

LEMMA 4.2. Suppose S is a densely defined symmetric (but not necessarily closed) operator in a Hilbert space (H, ( , )) and S is bounded below. Assume B is a symmetric operator satisfying S cr B a SF. Then

BF = SF.

(4.1)

Proof. This lemma was stated by Baxley [2] without proof. Since it was not immediately obvious to us and we have not seen a proof anywhere, we give one here for the sake of completeness. We have SczBczSF = SFczB*cS*.

(4.2)

Let y G D(SF). Then there exists a sequence yn in D(S) such that yn—>y in H and (S(yn-ym), yn-ym)->0 as n, m-»oo. Then yn belongs to D(B) and (B(yH-ym),yn-ym)-*0 (/i,m-K»). Furthermore, by (4.2), y e D(B*). Therefore, by the definition of BF, yeD{BF). Hence D(SF) c D(BF) c D(B*) and, by (4.2) again SF = >S'*|O(5,r)C: S* | D(BF)

=

B*\D(BF)

= BF.

(4-3)

Since SF and BF are self-adjoint, (4.3) implies that SF = BF. THEOREM 4.1. / / M is in the limit-circle case at a and if My = kay is non-oscillatory at a for some real ka, then My = ky is non-oscillatory at a for all real A.

Proof. By Corollary 3.3, there exists a positive function v e GP(I) such that the transformed operator N:=TVMTZX is pseudo-regular at a. Now Remark 3.2 shows that Nz = kz is non-oscillatory at a for every real A. Applying Lemma 3.4(2), we obtain that My = ky is non-oscillatory at a for every real A. In Corollary 2.1 it has been stated that My = ky is non-oscillatory at a and at b for all A < a if Mo is bounded below with lower bound a. The converse assertion also holds true: THEOREM 4.2. / / My = ky is non-oscillatory at a for some real ka and at b for some real kb, then MQ is bounded below and the domain of its Friedrichs extension MF is determined by

D(MF) = \yeD(M)\ I

lim ^ \ = lim ^ \ = o). x-*aVa(X)

x-*bV/,{X)

(4.4)

J

Here va and vb are arbitrary non-principal solutions of My = kay at a and of My = kby at b, respectively. Furthermore, the limits in (4.4) exist and arefinitefor every y e D(M) and L2(a, a) and (y/p)vb(^-)

e L2(p, b) for y e D(M), (4.5)

wb/

if va and vb do not vanish on (a, a] and [0, b), respectively. If ya and yb are any real solutions of My = kay and My = Afe.y, which have no zero in (a, a] and [j8, b), respectively, then l

) zL\a,a)

ya/

and (yJp)yb{L) Vfr/

€ L2(0, b) foryeD(MF).

(4.6)

SINGULAR STURM-LIOUVILLE PROBLEMS

559

Proof We may suppose that a < j3 and that ya and yb are positive on (a, a] and [)3, b), respectively. Then, by Lemma 3.6, there exists a positive function v e Gp(l) which coincides with ya on (a, a] and with yb on [j3, b). Let N:=TvMTZl be the transformed operator. Lemma 3.7(1) shows that NQ is bounded below. Thus, by Corollary 3.1, applied to Mx := MQ and Nt:=No, MQ is bounded below. Furthermore, Lemma 3.7(2) and Corollary 3.1 imply that for yeD(MF), (y/pMy/v)' = (y/P)(Tvy)' 6 L\I).

(4.7)

Since v coincides with ya on {a, a] and with yb on [0, b), this proves (4.6). If ya = va and yb — vb, then by Lemma 3.7(4) and Corollary 3.1, (4.4) holds true: D{MF) = TZlD{NF) = \ye D{M)\ lim Tvy(x) = lim Tvy(x) = ol. I

x—*a

x—*b

J

Furthermore, by Lemma 3.7(3), for all y e D(M), (4.7) is fulfilled and lim^

lim(rvy)(*) and iim^\ iim^\ = Mm (Tvy)(x)

x-*aVa{X)

x-*a

bV(X) x->bV b(X)

b

exist and are finite. THEOREM 4.3. Let My = Xy be non-oscillatory at a for some real X and let ua and va be principal and non-principal solutions of My = Xy at a which do not vanish in {a, oc\ Then, for every y 6 D(M), the following assertions are equivalent:

(4.8)

Km ^ 4 = 0, x-*a

Va(x) Wfl (^)

eL\a,(x),

(4.9)

eL\a,(x).

(4.10)

If M is also in the limit-circle case at a, then for every yeD(M) uniquely defined c, d e C such that (x) + dva(x)

forx^a,

there exist (4.11)

and the following assertions are equivalent to (4.8): [y,ua]M(a) = 0, y(x) = (c + o(l))ua(x) y = O(ua)

for x—»a and some c eC,

in (a, a].

(4.12) (4-13) (4.14)

If M is regular at a, then (4.8) is equivalent to y(a) = 0.

(4.15)

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H.-D. NIESSEN AND A. ZETTL

Proof. We may assume that M is regular at b. For otherwise we consider M on {a, b') for some b' e(a, b). This is possible by Lemma 2.3. Then My = Xy is non-oscillatory at b, too, and we can choose /? e (a, b) such that va has no zero in [/3, b). Then, by Lemma 2.3, to every y e D(M) there exists z e D(M) which coincides with y on (a, or] and which vanishes on [j8, 6). If y fulfills (4.8), then, by (4.4), z belongs to D{MF) and (4.6) applied to ya '= "a, yb •= va implies (4.9). With y := {pu'a)va - ua(pva)=t0, the identity (4.16) holds true. Since, by (4.5), (y/p)va(z/va)' eL2(a, a), (4.16) shows that (4.9) and (4.10) are equivalent for z and therefore for y, too. By Theorem 4.2, d:=\imx^a(z(x)/va(x)) exists and is finite. If (4.10) is fulfilled for y and hence for z, then Theorem 2.2(i) implies that d = 0, that is, (4.8) holds. Now let M be in the limit-circle case at a. Then ua and va belong to L2w(a, a). Therefore, by the variation-of-constants formula, there exist c, deC such that

ua(x)\ va{t)w{t)f{t)dt-va(x)\ Ja

ua(t)w(t)f(t) dt. Ja

Here, / := -y~\My - ky) e L2w{a, b) with Y\=(pu'a)va-ua{pv'a) (2.11), it follows that

eU.

From

i\va(t)\ fora Therefore,

Together with the fact that

this implies (4.11). Furthermore, c and d are uniquely defined since (4.11) and the equation lim*.^ {ua{x)lva(x)) = 0 imply that

To prove the equivalence of (4.8) and (4.12) we apply the transformation Tv, where v e GP(I) is a positive function which coincides with va on {a, a] (or with —va if va is negative on (a, a]). Such a function exists by Corollary 3.3. Then N := TvMTZl is pseudo-regular at a. Therefore, by Remark 3.2 and the facts that ua =£ 0 and lim(r«iO(x) = U m ^ T = 0, x-*a

x—*aVa\X)

SINGULAR STURM-LIOUVILLE PROBLEMS

561

we obtain

Then, since ua, va are real valued and ua belongs to D(M), we have, according to Lemma 3.2(8), [y, wo]w(«) = [Ty, Tua]N(a) = lim (Tvy)(x) lim (P(Tvua)')(x) - lim (P(Tvy)')(x) lim (Tvua)(x) x—*a

x—*a

x—*a

x—*a

= Mm (Tvy)(x) lim (P(Tvua)')(x). x—*a

x—*a

Hence, [y, ua]M(a) = 0 if and only if \imx^a(Tvy)(x) = 0, that is, if and only if

lim,_a (y(x)/va(x)) = 0. Since d = \imx^a(y(x)/va(x)) in (4.11), (4.8) is equivalent to (4.13). By (4.11) and since ua = o(va) for x—>a, (4.13) is equivalent to (4.14). Finally, if M is regular at a, then va(a) =£0 by Remark 2.2 and (4.8) reduces to y(a) = 0. COROLLARY 4.1. Suppose that the assumptions of Theorem 4.2 are fulfilled, let ua be a principal solution belonging to va, and let a el be chosen in such a way that ua and va do not vanish in (a, a]. Then the condition \imx_+a(y(x)/va(x)) = 0 in (4.4) can be replaced by the conditions (4.9) or (4.10). //, in addition, M is in the limit-circle case at a, then the condition lirn^o (y(x)/va(x)) — 0 in (4.4) can be replaced by any of the conditions (4.12) to (4.14). If M is in the limit-point case at a, then the condition \\mx_>a (y(x)/va(x)) = 0 in (4.4) is automatically fulfilled and may be dropped. A similar result holds for the right endpoint b.

Here the assertion concerning the limit-point case follows from the well-known fact [11, § 18.3] that at a limit-point endpoint no boundary condition is needed to determine self-adjoint extensions of Mo. An important feature of the characterization of the domain of the Friedrichs extension given by (4.8) or (4.9) or (4.10) is that it is independent of the limit-point/limit-circle classification. Thus it is not necessary first to determine this classification in order to impose the appropriate boundary conditions. The conditions (4.8) and (4.13)—in the limit-circle case and with stronger assumptions on the coefficients—are due to Rellich [13]. In the general case, (4.8) is due to Rosenberger [14], condition (4.9) has been obtained by Kalf [8], (4.10) seems to be new. Under more restrictive assumptions, Friedrichs [5] gives a condition which is slightly stronger than (4.9). Condition (4.15) shows that Theorem 4.2 reduces to the well-known fact that in the regular case the Friedrichs extension of the operator MQ (which then is always bounded below) is characterized by Dirichlet boundary conditions (see [12]). As Example 4 of [14] shows, in the limit-point case (4.8) is not equivalent to (4.13) and (4.14) (see Example I of § 6 with c = 6).

562

H.-D. NIESSEN AND A. ZETTL

The condition given by Kalf in [8] is weaker than (4.9) in so far as he assumes , <£ L2(a, a), uaMua ^ -y, \ua\2 vp)u instead of ua being a principal solution at a. In this case, instead of condition (1) of Lemma 3.5, we obtain by Lemma 3.2, QW~l ^ -ju on (a, a]. But this was all that was needed to prove Lemma 3.7(1), (2). Therefore, the technique used above also establishes this weaker condition. The following lemma gives a characterization of separated self-adjoint boundary conditions at a limit-circle endpoint at which My = Xy is non-oscillatory by means of principal and non-principal solutions. ua e GP(I) positive near a,

LEMMA 4.3. Suppose M is limit-circle at a and My = Xy is non-oscillatory at a for some (and hence any) X in U. Let ua and va be principal and non-principal solutions at a, respectively, for some fixed X inU. Then all (separated) self-adjoint boundary conditions for M at a are of the form

Al[y,ua]M(a) + A2[y,va]M(a) = 0 foryeD(M), where Ax, A2 are real and not both zero.

(4.17)

Proof. By Remark 3.1 we may assume that a is finite. Also we may assume that va is positive in (a, a) for some a in /. According to Corollary 3.3 there exists a positive function v e GP(I) which coincides with va near a. Let T :=TV and N := TVMT~X. Then by Corollary 3.3, N is regular at a. From Remark 2.2 we get

(Tua)(a) = 0, (P(Tua)')(a)±O. Furthermore, Tva is equal to 1 near a. Hence, (Tva)(a) = l,

(P(Tva)')(a) = 0.

From this and the fact that ua and va are real valued we obtain, for z e D(N), [z, Tua]N(a) = (P(Tua)')(a)z(a), [z, Tva]N(a) = ~(Pz')(a). (4.18) Now for Af all separated self-adjoint boundary conditions at a are of the form Cxz(a) + C2(Pz')(a) = 0 for z e D(N) with Cu C2 real, C\ + C\>0. By (4.18) and the non-singular real transformation (P(Tua)')(a) these boundary conditions can be written as A^z, Tua]N(a)+A2[z, Tva]N(a) = 0 for z in D(N) with A1} A2 real and not both zero. Using Lemma 3.2(8), we obtain the characterization (4.17). Next we characterize the Friedrichs extension of symmetric extensions of the minimal operator. This only makes sense if both endpoints are in the limit-circle case, for otherwise every non-trivial symmetric extension of Mo already is self-adjoint.

SINGULAR STURM-LIOUVILLE PROBLEMS

563

4.4. Suppose M is in the limit-circle case and My = ky is nonoscillatory at each endpoint a, b for some (and by Theorem 4.1 hence for every) real A. For any ka in U let ua and va be principal and non-principal solutions of My = kay at a. For any kb in IR let ub and vb be principal and non-principal solutions of My = Xby at b. Consider the boundary condition THEOREM

Bi[y, ub]M(b) + B2[y, vb]M(b) = 0, Bu B2 real, B2x + B22>0.

(4.19)

Define D(M ;Bl, B2) := {y e D(M)\ y satisfies (4.19) and is identically zero in a neighbourhood of a), M(BU B2)y := My for y e D(M ;BX, B2). Then M(BX, B2) is a symmetric operator in L2w(a, b) which is bounded below. Its Friedrichs extension has domain DF(M ; Bx, B2) = {y e D(M)\ y satisfies (4.19) and any one of the conditions(4.8), (4.9), (4.10), (4.12), (4.13) or (4.14)}. A similar result holds if the conditions at the endpoints are interchanged. REMARK 4.1. If the endpoint b is regular, then, as was seen in the proof of Lemma 4.3, (4.19) reduces to

Cxy(b) + C2{py'){b) = 0, with Cx, C2 real, C2 + C\>0. Theorem 4.4 with condition (4.13) for this case—and with stronger assumptions on the coefficients—is due to Rellich [13]. Our proof is different from his. Proof of Theorem 4.4. Using Remark 3.1 once more, we may suppose that (a, b)\s finite. Let v e GP(I) be a positive function which coincides with va near a and with vb near b. (We may suppose that va and vb are positive near a and b, respectively.) Let T:=TV, N:= TVMT~' and N(BU B2):= TVM(BU B2)T~\ Then Af is regular at a and at b. As has been shown in the proof of Lemma 4.3 (for a instead of b), there exist real C1} C2 which do not both vanish such that the domain of N(Blt B2) equals {z e D(N)\ Cxz{b) + C2(Pz')(b) = 0 and z is identically zero in a neighbourhood of a}. Then obviously N(Bl} B2) is symmetric and its closure N(BU B2) has domain {z e D(N)\ Cxz(b) + C2(Pz'){b) = 0 and z(a) = (Pz'){a) = 0}. By Theorem 2.2 of [12], N(Blt B2) is bounded below and the domain of its Friedrichs extension is determined by {z € D{N)\ Cxz{b) + C2(Pz'){b) = 0 and z(a) = 0}. Applying Lemma 3.2(7) and Corollary 3.1 with Mx := M(BX, B2), we obtain that M(BX, B2) is symmetric, bounded below and its Friedrichs extension has domain DF(M ; Bx, B2) = {ye D(M)\ y satisfies (4.19) and (4.8)}. Here by Theorem 4.3, we can replace (4.8) by any of the conditions (4.9), (4.10), (4.12), (4.13) and (4.14).

564

H.-D. NIESSEN AND A. ZETTL

5. Eigenvalue comparisons in the non-oscillatory limit-circle case Throughout this section we assume that M is in the limit-circle case and that My = ky is non-oscillatory at each endpoint a, b for some (and hence for any) real A. Let Xa and Xb be real. Then by Theorem 2.1 there exist principal and non-principal solutions ua and va of My = kay at a and principal and non-principal solutions ub and vb of My = Xby at b. We may assume that va and vb are positive near a and b, respectively. Then, by Lemma 3.6, there exists a positive function v e GP(I) which coincides with va near a and with vb near b. By Remark 3.1, we may suppose that a and b are finite. Then Lemma 3.5 implies that N := TVMT~X is regular at a and at b. Therefore, by Remark 2.2, (Tua)(a) = (Tub)(b) = 0,

(P(Tua)')(a)*0,

(P(Tub)')(b)±0.

(5.1)

Hence, on multiplying ua and ub by suitable non-zero real numbers, we may assume that -l.

(5.2)

Furthermore, since Tva and Tvb are equal to 1 near a and b, respectively, (Tva)(a) = (Tvb)(b) = 1, (P(Tva)')(a) = (P(Tvb)')(b) = 0.

(5.3)

It is well known [11, § 18.2], that in this regular case Nx is a self-adjoint extension of No if and only if its domain D(NX) consists of all z eD(N) which satisfy a certain 'self-adjoint' boundary condition )=0

-

(5 4)

'

Here A, B are (2 x 2)-matrices of complex numbers satisfying the following two conditions: rank(/l : B) = 2,

(5.5)

AJA* = BJB*, with 7 = ^ ~ J ) ,

(5.6)

where A : B denotes the (2 x 4)-matrix obtained by placing B to the right of A and A* denotes the adjoint (conjugate transpose) of A. For Ax

A2\

n

/0

0

(5.4) reduces to the important special case of 'separated' boundary conditions: AlZ(a)+A2(Pz')(a)

= 0,

BlZ(b) + B2(Pz')(b) = Q.

(5.7) (5.8)

Then (5.5), (5.6) reduce to (AuA2)*0, AXA2 = AXA2)

(B1}B2)*0,

(5.9)

BXB2 = BXB2.

If Axj^0, we may multiply (5.7) by AX. This does not change (5.7), but AXA2 = AXA2 becomes equivalent to Ax and A2 being real for the new coefficients Ax and A2. A similar result is obtained by multiplication with A2 if

SINGULAR STURM-LIOUVILLE PROBLEMS

565

A{ = 0. Similarly, (5.8) may be multiplied by Bx or by B2. Then the selfadjointness of (5.7) and (5.8) is equivalent to (5.9) and Au A2 and Bx, B2 are real.

(5.10)

It is sometimes convenient to reformulate (5.7), (5.8), (5.9), (5.10) as follows: cos a. z(a)- sin a. (Pz')(«) = 0, for 0 ^ a
(5.11)

cos j3. z{b) - sin p. (Pz')(b) = 0, for 0 < p ^ n.

(5.12)

Another important special case of the self-adjoint boundary conditions (5.4) considered by Weidmann [15] is ~

- c " 1 )) '

where0=?tce

This case includes for c = l the periodic, and for c = - l the anti-periodic boundary conditions. All the above-mentioned boundary conditions can be reformulated using ua, va, ub, vb as follows. By (5.1), (5.2) and (5.3) we obtain, for de{a, b) and zeD(N), [z, Tud]N{d) = ~z(d),

[z, Tvd]N(d) = -(Pz')(d).

(5.14)

Therefore, the boundary condition (5.4) can be written as z, Tva]N{a))

\[z,Tvb]N(b)

and the conditions (5.5), (5.6) for self-adjointness remain unchanged. Similarly, the other boundary conditions may be reformulated: the separated boundary conditions (5.7), (5.8) are equivalent to A\z, Tua]N{a) + A2[z, Tva]N(a) = 0, Bx[z, Tub]N{b) + B2[z, Tvb]N(b) = 0,

(5.16) (5.17)

where Ax, A2, Blt B2 are real numbers such that (Au A2)=£Q, (Bx, B The normalized form (5.11), (5.12) of the separated boundary conditions give rise to cos a. [z, Tua]N(a) - sin a. [z, Tva]N(a) = 0, with 0 «s a < JZ, cos p. [z, Tub]N(b) - sin p. [z, Tvb]N(b) = 0, with 0 < p ^ x.

(5.18) (5.19)

Since, by Lemma 3.2(8), [Ty, Tud]N{d) = [y, ud]M(d), [Ty, Tvd]N(d) = [y, ^ ] M ( ^ ) > we have proved the following characterization of self-adjoint extensions of Mo: THEOREM 5.1. The operator Mxis a self-adjoint extension of Mo if and only if its domain consists of all y e D(M) which satisfy the self-adjoint boundary conditions

[y, vb]M(b)/ where the (2 x 2)-matrices A and B fulfill the self-adjointness conditions (5.5), (5.6).

566

H.-D. NIESSEN AND A. ZETTL

Special cases of self-adjoint boundary conditions are the 'generalized periodic' boundary conditions (5.20) with A and B as in (5.13) and the separated boundary conditions A\[y, ua]M(a) + A2[y, va]M(a) = 0, B1[y, ub]M{b) + B2[y, vb]M{b) = 0,

(5.21) (5.22)

where Ax, A2, Bl} B2 are real numbers such that (AX) A2)^0 a n d (Bx, B2)^0. In normalized form, these separated boundary conditions can be stated as cos a. [y, ua]M{a) - sin a. [y, va]M(a) = 0, where 0 *£ oc < n,

(5.23)

cos $. [y, ub]M{b) - sin j8. [y, vb]M{b) = 0, where 0 < /? *s n.

(5.24)

REMARK 5.1. Since [y,ud]M{d) =-{Ty){d), [y,vd]M{d) =-{P{Ty)'){d), Dirichlet condition z{d) = 0 for N is transformed to

[y,ud]M(d) = 0

the (5.25)

for M and the Neumann condition (Pz')(d) = 0 for Af is transformed to [y,vd]M(d) = 0.

(5.26)

Thus (5.25), (5.26) are the singular analogues of the Dirichlet and Neumann conditions, respectively. Also, (5.21) and (5.22) or, equivalently, (5.23) and (5.24) are the general singular separated self-adjoint boundary conditions. The next four theorems are all proved by reduction to the regular case with the transformation Tv. THEOREM 5.2. (1) The spectrum of the self-adjoint extension Mx of Mo defined by (5.20), (5.5), (5.6) is discrete and bounded below. It therefore consists of an infinite sequence of eigenvalues Xn (n e N) tending to infinity which can be ordered so that

-oo
(5.27)

Here the eigenvalues are counted according to their multiplicity. Each eigenvalue can have multiplicity at most 2. Therefore, in (5.27) at most two consecutive eigenvalues can be equal. (2) Let n has at least n — 2 and at most n zeros in (a, b). If \imx^a(q)n(x)/va(x)) = 0 then q>n has at most n — \ zeros in {a, b). If kn = kn+xis an eigenvalue of multiplicity 2, then every real eigenfunction of kn has either n — \ or n zeros in {a, b). There exist real eigenfunctions q)n and q>n+x of Xn = Xn+X with n — \ and n zeros in {a, b), respectively. (3) Let N(X) denote the number of eigenvalues of Mx in (—°°, A]. Then VA

jzia M\p(x)J

and

nV

dxe(O, oo) (A(>0)-*oo)

(f v e - r <—)•

(5.28)

SINGULAR STURM-LIOUVILLE PROBLEMS

567

Note that the limit in (2) exists by Theorem 4.2. By (5.1), (5.2) and Lemma 3.2(8), this is equivalent to [cpn, ua]M(a) - 0. Proof. (1) The self-adjoint extension Nx := TvMxT~l of No is defined by the boundary conditions (5.4), (5.5), (5.6). Since N is regular, the spectrum of Af, is bounded below and consists of an infinite sequence of eigenvalues A,, (n e N) tending to infinity and which can be ordered so that (5.27) holds true. Furthermore, each eigenvalue can have multiplicity at most 2. (2) According to [15, Theorem 13.5], every real eigenfunction tpn of Af, belonging to kn has at least n — 2 zeros in (a, b) and at most n zeros in [a, b). If Xn = Xn+x, then every real eigenfunction of kn has either n —• 1 or n zeros in (a, b) and there exist real eigenfunctions tpn and i//,,+1 belonging to An with n — 1 and n zeros in (a, b), respectively. Now the spectra of Mx and A^ are identical and ipn is an eigenfunction of Nx if and only if (pn := T~lil)n is an eigenfunction of Mx. Furthermore, the zeros of q)n and ipn which lie in (a, b) are the same and

Therefore, a is a zero of ipn if and only if lim^^a (q>n(x)/va(x)) = 0. Together with the results for ipn stated above, this proves (2). (3) Let A^(A) denote the number of zeros in [a, b) of the solution z of Nz = Xz with z{a) = 0. Then by [1, Lemma 2.1,a)], VA Here, V(W/P) = y/(w/p) is integrable since N is regular (or, since M is non-oscillatory and in the limit-circle case at both endpoints; see Corollary 3.2). Also tyn has at least n — 2 zeros in (a, b) and i/;n+1 has at most n + 1 zeros in [a, b). Therefore, by Sturm's Separation Theorem and since the eigenvalues of Nx and Mi are identical, n — 2 =*£ A^(A) =ss n + 2 for every Ae[An, kn+i), that is, N(k)-2^Ni(k)^N(k) + 2. Hence (5.30) implies (5.28). Finally, (5.29) follows from (5.28) since N(kn) = n and An-»<» («-»<»). To make comparisons between the eigenvalues for different boundary conditions easier we introduce the following notation. Let kn(a, P) denote the eigenvalues of the self-adjoint extension of M'o defined by the normalized separated boundary conditions (5.23), (5.24), and let kn(c) denote those of the extension defined by the 'generalized periodic' boundary conditions (5.20), (5.13), for neN. By Theorem 5.2 we may suppose that the sequences kn{<x,ft)and kn(c) are ordered so that they are increasing in n. THEOREM 5.3. Each eigenvalue kn{a, ft) is simple. If q>n is an eigenfunction belonging to kn{a, P), then (pn has exactly n — \ zeros in {a, b), where neN. Also, for neN, (1) kn(a, P) is a continuous function of a, P, (2) kn(a, P) is strictly decreasing in a for each fixed P and strictly increasing in P for eachfixedoc,

568

H.-D. NIESSEN AND A. ZETTL

(3) with the understanding that terms involving A_1} Ao are not present we have, for ae [0, x), f$e (0, it] and neN,

Proof. Let M, denote the self-adjoint extension of Mo defined by (5.23), (5.24). Then the self-adjoint extension NX:=TVMXTZX of No is defined by means of the boundary conditions (5.11), (5.12). It has the same spectrum as M, and \pn := Tvcpn is an eigenfunction of Nx corresponding to the eigenvalue Xn{a, /?). Then, since N is regular, Theorem 13.2 of [15] shows that xpn, and hence q>n, has exactly n - 1 zeros in (a, b). Now (1), (2), (3) follow from [7,1, § 5, Theorem 3] applied to Nx. (The notation in [7] differs slightly from ours. In our notation, Xn(a, /?) is increasing in jS and /3 = 0 has to be replaced by /? = n in (3). Also, the first two inequalities in (3) can be proved to be strict.) THEOREM 5.4. Let 0, then

c)^... .

(5.31)

Then cpx has no zero in {a, b) and x^a

Va(x)

x^b

Vb(x)

For cp :=• q>2n('> c) or cp := q>2n+i('> c) there exist exactly 2n zeros in (a, b) if \imx^,a(q)(x)/va(x)) i= 0 and 2n — l zeros in (a, b) if lini,.^, (q?(x)/va(x)) = 0. (2) Ifc<0, then Xt(c) ^ A2(c) < A3(c) ^ A4(c) < ... .

(5.32)

For cp := (pzn-i(-, c) or cp := c) there exist exactly 2n — 1 zeros in (a, b) if \imx^a((p(x)/va(x)) =£0 and In-2 zeros in (a, b) if \\mx^a((p(x)lva{x)) = 0. Proof. Let Mx denote the self-adjoint extension of Mo defined by the boundary conditions (5.20), (5.13). Then the self-adjoint extension M, := TvMiT~l of No is defined by means of (5.4), (5.13). It has the same spectrum as Mx and il>n(', c) := Tvcpn(-, c) is an eigenfunction of Nx corresponding to the eigenvalue Xn(c).

Since N is regular, Theorem 13.7 and Theorem 13.8 of [15] can be applied to ty. This shows that the orderings (5.31), (5.32) hold true and that, in Case (1), \l>\(', c) has no zero in [a, b] and ip2n(', c) a n d V2,i+i('> c) both have exactly 2n zeros in [a, b). In Case (2), ip2n-i('> c) a n dtyzn{%>c) both have exactly 2n — 1 zeros in [a, b). Now the zeros of ipn(', c) and cpn{-, c) in (a, b) are the same, and ipn(a, c) = 0 is equivalent to lim ((pn(x, c)lva{x)) = 0.

SINGULAR STURM-LIOUVILLE PROBLEMS

569

Therefore, the remaining statements of (1) and (2) follow from the assertions on \\>n stated above. The next result compares eigenvalues of generalized periodic boundary conditions with those of Dirichlet and Neumann boundary conditions. THEOREM

5.5. For any c > 0 we have

A2«(0, TC) Proof. Since N := TvMT~l is regular and the eigenvalues of any self-adjoint extension A/j of Mo are the same as the eigenvalues of the self-adjoint extension Nii=TvM^T~l of N0:=TVMQTZX, Theorem 5.5 follows immediately from [15, Theorem 13.10] applied to N. 6. Examples EXAMPLE

I (Bessel's equation). We consider the following form of Bessel's

equation: My := — y" + —?y = Ay, withO<x=£l. (6.1) x Clearly M is regular at 1 and for 0=£celR singular at 0. In particular, (6.1) is non-oscillatory at 1 for all real A. Let v := V(c + \) 1+v

with Re v 5= 0.

v

Then for A = 0, x and x*~ are solutions. These are linearly independent except when c = — \. In that case, x1 and xMog* are linearly independent solutions. Therefore, (6.1) is in the limit-circle case at 0 if and only if c < 4. For A = 0, (6.1) is non-oscillatory at 0 if and only if v is real, that is, if and only if c 2= - 4. Then principal and non-principal solutions u0 and v0 for A = 0 at 0 are v

o(x)

^~v

for c > —\,

logx

for c = -%.

= 1 1,

,

1

We distinguish three cases. Case 1: c^\. Then M is in the limit-point case at 0. Hence no boundary condition is required at 0 to obtain self-adjoint realizations of M in L2(0, 1). In particular, by Theorem 4.2 and condition (4.15) of Theorem 4.3, D(MF) = {yeD(M)\

570

H.-D. NIESSEN AND A. ZETTL

Case 2: - J ^ c < \. By Theorem 4.2, MQ is bounded below. Since M is in the limit-circle case at 0 in this case, a boundary condition is needed to obtain self-adjoint realizations of M. Therefore, by Theorem 4.2, Corollary 4.1, and condition (4.15) of Theorem 4.3 again, the domain of the Friedrichs extension MF of Mo is given by D(MF) = {ye D(M)\ y(l) = 0 and lim [(J + v)y(x)xv^

- xv^y'(x)]

= 0}.

:r-+0

Here condition (4.12), for b instead of a, has been used to formulate the boundary condition at 0. By Corollary 4.1, we can use the analogues for the right endpoint of the conditions (4.8), (4.9), (4.10), (4.13) and (4.14) as well. In our concrete situation they can be formulated as follows: (4.8):

lim JCV"^(JC) = 0

for v > 0,

lim (x* IOPx}~ v(x} = 0 for v = 0*

(4.9):

y'-(^ + v)x-1yeL2(0, 1);

(4.10): x~ly e L2(0, 1) l

for v > 0, 2

(x log x)~ y e L (0, 1)

for v = 0;

v 5

(4.13): lim*~ " >'(x) exists and is finite; (4.14): x-y^y

e L°°(0,1).

It should be remarked that for y e D(M) the following limits exist: l

y(x)

forv>0,

lim (x^ \ogx)~1y(x)

for v = 0.

x->0

This follows from Theorem 4.2. In particular, for 0«sv<^ and

yeD(M),

We may also consider the Friedrichs extensions of the symmetric extensions M(AU i42) and M(BU B2) of M^ defined by D(M ;AuA2):={ye M(AU

D(M)\ Ax[y, uo]M(0) + A2[y, vo]M(0) = 0andy is

identically zero in a neighbourhood of 1}, A2)y := My for y € D(M ; Au A2),

and D(M ; Bx, B2) := {y e D(M)\ Biy(l)

+ B2y'(l) = 0 and y is identically zero in a neighbourhood of 0},

M(BuB2)y=My

for y e D(M ; Bu B2),

respectively. Here Ax, A2, Bx, B2 are real numbers such that {Ax, (Bx, B2)¥^0. According to Theorem 4.4, condition (4.15) of Theorem 4.3, Remark 4.1, and Corollary 4.1, the Friedrichs extensions of these operators have

SINGULAR STURM-LIOUVILLE PROBLEMS

571

domain DF(M ;AuA2):={ye

D(M)\ Ax[y, uo]M(0) + A2[y, vo]M(0) = 0, y(l) = 0}

and DF{M ; Bx, B2) := \y e D(M)\ Bxy(l) + B2y'{\) = 0 and lim [(5 + v)y(x)xv' x-*0

Here again the boundary condition at 0, which stems from (4.12), may be replaced by those coming from (4.8), (4.9), (4.10), (4.13) or (4.14) formulated above for the example considered here. Case 3: c< — \. Since (6.1) is oscillatory at 0 for A = 0, the minimal operator Mo is not bounded below by Corollary 2.1 and Theorem 4.1 and thus there is no Friedrichs extension in this case. EXAMPLE II (Legendre's equation). We consider the following form of Legendre's equation:

My=-{py')' = Xy, p{x):=\-x2, where -\<x<\.

(6.2)

Here both endpoints are singular. For A = 0, 1 -I- r

u(x) := 1, v(x) = log

1-x

are linearly independent solutions of (6.2). They belong to L2(—1, 1) and are non-oscillatory at both endpoints. Clearly u is a principal and v a non-principal solution at both endpoints. Hence, by Theorem 4.2 and Corollary 4.1, the preminimal operator MQ is bounded below and the domain of its Friedrichs extension can be characterized as follows: D{MF) = {yeD\ (py')(-l) = (py')(l) = 0}. (6.3) Here again condition (4.12), for both endpoints, has been used to formulate the boundary conditions. By Corollary 4.1, we can use the conditions (4.8), (4.9), (4.10), (4.13) or (4.14) instead of (4.12). These conditions can be formulated here as follows: (4.8):

v(x) lim y\ ' = 0 ; *-±ilog(l TJC)

(4.9):

(4.13):

lim y(x) exists and is finite;

(4.14): 3>eL°°(-l, 1). Again, Theorem 4.2 shows that for y e D(M) the limits limx_»±1 (.y(*)/log (1 exist and are finite.

572

H.-D. NIESSEN AND A. ZETTL

The general self-adjoint boundary conditions (5.20) for (6.2) can be formulated as follows:

where the (2 x 2)-matrices A and B fulfill the self-adjointness conditions (5.5), (5.6). Therefore, by Theorem 5.1, every self-adjoint extension of the minimal operator Mo is characterized by self-adjoint boundary conditions (6.4) with suitable matrices A and B. The special case of self-adjoint separated boundary conditions takes on the form Mb, u]M(-l)+A2[y, v]M(-l) = 0, %«W1)

+ %»W1)

= O,

(6.5) (6.6)

where Ax, A2, Bx, B2 are real numbers such that (Au A2) =£0 and (/?,, B2)¥^0. When Ax = Bl — l, A2 = B2 = 0, we have the Friedrichs extension of M0. It is interesting to note that the Friedrichs extension is determined by the boundary conditions

which have the appearance of the Neumann boundary conditions. But they are, in fact, the analogue of the Dirichlet conditions in the regular case; see Remark 5.1. It is also noteworthy that the Friedrichs extension of the Legendre expression M on the interval (—1,1) cannot be approximated by the Friedrichs extension of M on ( - 1 + e, 1 - e) since the latter is determined by the boundary conditions y(-l + e) = 0 = y(l - e) for each e, with 0 < e< 1. 7. Proof of Lemma 2.3 LEMMA 7.1. Let a< a0 be real constants. For s e [—£, £] denote by ys the (real) solution of the initial value problem

My = 0 on [a, j8], y(a) = A,

(py')(a) = h+s,

and suppose that h +s — £ ^ (pys)(0 *^h + s + e for t e [a, j3] and s e [—e, e\. Then there exists s e [-£, £] such that (a) ys(B) = A + h

-=:B.

Furthermore, (P 1 [p 1 (b) min(,4, B)-2e \ - ^ ys(x) ^ max(,4, B) + 2e I - for x e [a, 0], Ja p Ja p pv; 2 ^(|fc| + 2£)2

(c) Ja

-. Ja P

(7.1)

SINGULAR STURM-LIOUVILLE PROBLEMS

573

Proof. Dividing (7.1) by p(t) and integrating over [oc, x] yields, for x e [a, ft],

A + (h+s-s) lX-^-^ys(x)^A + (h+s + e) [*-£-. jp{t)

(7.2)

Kp{t)

For x = ft this implies that

and especially we get

Since ys(fi) depends continuously on s, there exists se[-e,e]

such that

Now from (7.2) we get for h 2* 0, that is, for B ^ A, iin(,4, B) - 2e\ min(A,B)-2ei-^= A-2ef

*

dt Kp{t)

and for h <0, that is, for B
Pit) = max(A, B) + 2e I

Pit)

This proves (b). Part (c) follows by (7.1) from the inequalities

f

oe

P

J<xP

LEMMA 7.2. Let a 0 be real constants. Then there exists a real-valued function y e D(M, [a, ft]) such that

(a) y{a) = 0, (b) \y{x)\*e (c)

forxe[a,p],

574

H.-D. NIESSEN AND A. ZETTL

Proof. Denote by yx, y2 the (real) solutions of My = 0 which satisfy y\{oc) = \,

{py[){oc) = 0, y2(a) = 0, (pyi)(a) = l.

(7.3)

We may suppose that yx and y2 are both positive on (a, /ft). Otherwise, by (7.3), we can find ft' e {a, /ft] such that yx and y2 are positive on (a, /?'). Applying the lemma to [a, /ft'], we then obtain a function u e D(M, [a, /ft']) with the properties (a)-(c) listed above with /3 replaced by jft'. Then the function y defined by u(x) .0

forxe[ar,j8'], for x e (/?', f$]

fulfills the assertions of the lemma. Now let

Then Y{cc) = E2) det Y(x) = det Y(a) = 1 and

Let x e {a, p) and define -i

Since y2 is a principal solution and yx is a non-principal solution with respect to the left endpoint a of the interval (a, )3), we get from (2.11) (for (a, j8) instead of

"V

Mll '

(T->a).

(7.4)

*(T)

Let \x

for a «£ t *s

T,

iT for x < t ^ )3, where the real constants gT and hT are defined by - 1 / r/3

(7.5) )

•

(7.6)

Here the denominators are positive by (2.11) and (7.4). Then ).

(7.7)

(All O's and o's in this proof are for x—* a and, later on, uniformly with respect tOJC.)

Now on [a, /3], let uT be the solution of -{pu'xy + qux = wfx, 2

uT(P) = {pulXfi) = 0.

Then ux e GJ[a, p]) and Mux =fz e L w(a, 0). Thus, uT belongs to D(M, [a, j8]).

SINGULAR STURM-LIOUVILLE PROBLEMS

575

By the variatiori-of-constants formula, we get

A straightforward calculation using (7.5), (7.6), (7.8) and Y(a) = E2 shows that uT{a) = 0,

{pu'x)(oc)=A'.

Furthermore, since w e L(a, /3), we get from (7.8) and (7.7), {) {\\) puxl For a^x < r, (7.8) shows that

{p{))

p

(7.9)

Here the last term is once more O{q){x)). By (7.7), the first term is 0(1) and its first component is O(q)(r)). The second term therefore is O(\Y(x) — E2\), that is, o(l). Together with (7.4) and (7.9), this shows that KlU^l),

\\puT\\a> = O(l),

c T := sup \(puT)(x)\ = o(l).

Therefore, re

rp(Du'}2

rTi

cpi

[ pu;* = f i£M.« ! || |< || i [ i + cj f I _ o Jo-

Joe

P

Ja P

(t_»ff).

Jx P

Hence for T sufficiently close to a, y := uT fulfills the assertions of the lemma. REMARK

7.1. A similar argument shows that Lemma 7.2 holds with (a)

replaced by

(a') y(a) = 0, y(fi) = O, (/>/)(*) = 0, Proof of Lemma 2.3. The idea of the proof is the following. If Lemma 7.1 is applicable to the whole interval [a, )8], it shows the existence of a solution y of My = 0 which has the correct values A and B at the endpoints a and /? respectively, and which fulfills Conditions (b) and (c) of Lemma 2.3. Then Lemma 7.2 and Remark 7.1 are used to correct y in such a way that the values of py' at a and jS are now A' and B' respectively. This correction does not destroy properties yeD(M, [a, /?]), (b), (c) of Lemma 2.3 (except of increasing e). If Lemma 7.1 is not applicable to the interval [a, ft], the latter will be partitioned into subintervals to which Lemma 7.1 is applicable: define h:=(B-A)(\ , \Ja pi choose r) e (0, 1] such that -^s, P

(7.10)

576

H.-D. N1ESSEN AND A. ZETTL

and let yu y2 be linearly independent solutions of My = 0 on [a, /?]. Then Y:= is invertible and uniformly continuous on [a, /?]. Therefore, there exists 6 > 0 such that for all real c and d, with \c\ ^ max(|i4|, \B\), \d\ ^ \h\ + 1, and for all xltx2€ [a, p] with |*i - x 2 \ =s 6, (7.11) where || || denotes the Euclidean norm in C2. Let n e N be such that (ft — a) In =£ 6 and define i

Ca' 1 \ Ja p

B- a

Then a0 = a, ocn = ft, Ao-A, A n = B. W e apply L e m m a 7.1 t o each of the intervals /, : = [<xh ar, +1 ] (/ = 0 , . . . , n - 1) with Aj instead of A and t] instead of e. Denote by yiiS the solution of My =0 on /, with _y(flr,) = Ah

{py'){(x,) = h+s (s e[-rj, rj]).

Then, for telh

<<^)={ h A ; s )

yt.s

pyl

and since |y4,| =^ max (|>l|, \B\), by (7.11), we have

, and ai+l - a^ 6, and

h + s — rj =5 (pyls)(t) ^h +s + T) for t e /, and s e [-rj, rj]. Hence the assumptions of Lemma 7.1 are fulfilled and there exists Sje[-rj, rj] such that y( :=y,,J(. fulfills the following conditions: yi{ai) = Ah

yi(ai+1) = At + h

- = Ai+l, J«i

in(Ah Ai+l) - It] mm

(7.12)

P

-=sy,(x)^max(>l l , Ai+1) + 2rj J<x, P

Ja,

X

~_-

(xelt),

(7.13)

P

(714)

Now we apply Lemma 7.2 and Remark 7.1 to each 7, with r/2 j£;+l {lip) instead of e and obtain real-valued functions M, and vt with uh Vi e D(M, Ii),

(7.15)

u^)

(7.16)

= Viiai) = Ui(ai+l) = V;(ai+l) = 0,

= 0, (7.17)

SINGULAR STURM-LIOUVILLE PROBLEMS

\v,(x)\ =£ rj2 f

Mx)\,

J

Oi

-

577

for x e /„

(7.19)

P

pu'2,\ pv'^r,2 -. (7.20) We now define y(x) := )>,(*) + w,(x) + i>,(jt) for xelit i = 0,..., n - 1. Then by (7.12) and (7.16), y is continuous at or, (i = 1,..., n - 1) and y(a) = A, _y(/3) = B\ by (7.17) and (7.18), py' is continuous at or, (i = 1, ..., n - 1) and (p^')(ar) = A', (py')(($) = B'. Since _y, is a solution of My=0 on /,, (7.15) then implies that yeD(M, [a, )8]). Furthermore, (7.13), (7.19), (7.10) and min04, B)^ min(y4,, Ai+l), max(i4,, Ai+1) ^ max(A, B) imply (b). By Minkowski's inequality, (7.14) and (7.20) we obtain

0

•<*• + !

that is, ( Jai

)

P

Summing up these inequalities and using (7.10), we get assertion (c): Ja

py'2^\h\2\

- + (16r,2 + Sri\h\)\ -^\B-A\2(\ Ja p Ja p

\Ja p)

+ e.

Acknowledgement A. Zettl thanks the Department of Mathematics, University of Essen for its generous hospitality during the spring of 1989 when this work was done. References 1. F. V. ATKINSON and A. B. MINGARELLI, 'Asymptotics of the number of zeros and of the eigenvalues of general weighted Sturm-Liouville problems', J. reine angew. Math. 375/376 (1987) 380-393. 2. J. V. BAXLEY, 'Eigenvalues of singular differential operators by finite difference methods I', J. Math. Anal. Appl. 37 (1972) 244-254. 3. W. N. EVERITT, 'On the transformation theory of ordinary second-order linear symmetric differential expressions', Czechoslovak Math. J. 32 (1982) 275-306. 4. K. FRIEDRICHS, 'Spektraltheorie halbbeschrankter Operatoren und Anwendungen auf die Spektralzerlegung von Differentialoperatoren I, II', Math. Ann. 109 (1933/34) 465-487. Corrections, ibid. 110 (1934/35) 777-779. 5. K. FRIEDRICHS, 'Uber die ausgezeichnete Randbedingung in der Spektraltheorie der halbbeschrankten gewohnlichen Differentialoperatoren zweiter Ordnung', Math. Ann. 112 (1935/36) 1-23. 6. P. HARTMAN, Ordinary differential equations (Wiley, New York, 1964). 7. K. JORGENS, Spectral theory of second-order ordinary differential operators (Aarhus Universitet, 1962/63).

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SINGULAR STURM-LIOUVILLE PROBLEMS

8. H. KALF, 'A characterization of the Friedrichs extension of Sturm-Liouville operators', J. London Math. Soc. (2) 17 (1978) 511-521. 9. M. K. KWONG and A. ZETTL, 'Weighted norm inequalities of sum form involving derivatives', Proc. Roy. Soc. Edinburgh Sect. A 88 (1981) 121-134. 10. W. LEIGHTON, 'Principal quadratic functional', Trans. Amer. Math. Soc. 67 (1949) 253-274. 11. M. A. NAIMARK, Linear differential operators, vol. II (Ungar, New York, 1968). 12. H. D. NIESSEN and A. ZETTL, 'The Friedrichs extension of regular differential operators', Proc. Roy. Soc. Edinburgh Sect. A 114 (1990) 229-236. 13. F. RELLICH, 'Halbbeschrankte gewohnliche Differentialoperatoren zweiter Ordnung', Math. Ann. 122 (1950/51) 343-368. 14. R. ROSENBERGER, 'Charakterisierung der Friedrichsfortsetzung von haibbeschrankten SturmLiouville Operatoren', Dissertation, Technische Hochschule Darmstadt, 1984. 15. J. WEIDMANN, Spectral theory of ordinary differential operators, Lecture Notes in Mathematics 1258 (Springer, Berlin, 1987). Department of Mathematics University of Essen Universitdtsstr. 3 D-4300 Essen 1

Germany

Department of Mathematical Sciences Northern Illinois University DeKalb Illinois 60115-2888

U.S.A.