Spectral Theory of Differential Operators (North-Holland Mathematics Studies, 55)

NORTH. HOLLAND

MATHEMATICS STUDIES

Spectra I Theory of Differential Operators

I.W. KNOWLES R.T.LEWlS Editors

NORTH·HOlLAND

55

SPECTRAL THEORY OF DIFFERENTIAL OPERATORS

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© North-Holland

Publishing Company,1981

All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, ortransmitted, in any form or by.any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner.

ISBN: 0444 86277 3

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PRINTED IN THE NETHERLANDS

This volume is respectfully dedicated to Professor F.V. Atkinson on the ocassion of his sixty-fifth birthday.

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NORTH-HOLLAND MATHEMATICS STUDIES

Spectral Theory of Differential Operators Proceedings ofthe Conference held at the University of Alabama in Birmingham, Birmingham,Alabama, U.S.A., March 26-28, 1981

Edited by

IAN W. KNOWLES and

ROGER T. LEWIS University of Alabama Birmingham, Alabama, U.S.A.

19]1

N.H 1981

q~c

NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM. NEW YORK. OXFORD

55

PREFACE This volume forms a permanent record of lectures given at the International Conference on Spectral Theory of Differential Operators held at the University of Alabama in Birmingham March 26-28, 1981. The conference was supported by about 90 mathematicians from North America and Europe. Its main purpose was to provide a forum for the discussion of recent work in certain areas of the theory of ordinary and partial differential equations loosely connected under the general heading of Spectral Theory. Invited one-hour plenary lectures were given by F. V. Atkinson, who gave a series of three lectures, P. Deift, \~. N. Everitt, H. Ka If, T. Kato, R. M. Kauffman, M. Schechter and B. Simon. The remainder of the programme consisted of invited special session lectures, each of one-half hour duration. On behalf of the participants, the conference directors acknowledge, with gratitude, the generous financial support provided by the School of Natural Sciences and Mathematics and the School of Graduate Studies of the University of Alabama in Birmingham. \~e are especially grateful to Professor Peter V. O'Neil, Chairman of the Department of Mathematics, for his support and encouragement. Without this support the conference could not have taken place. We acknowledge also the valuable support provided by the faculty and staff of the Department of Mathematics. Here, we are particularly grateful to Professor Fred Martens, for his efficient direction of the local arrangements, and to Mrs. Eileen Schauer for her speedy and expert typing of much of the conference material, including many of the articles appearing in this volume. Finally, it is a pleasure to acknowledge the friendly assistance of Drs. Arjen Sevenster, Editor of the Mathematics Studies Series of North-Holland, during the preparation of these Proceedings. Ian \1. Knowles Roger T. Lewi s Conference Directors

vii

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CONTENTS C. D. Ahlbrandt, D. B. Hinton and R. T. Lewis Transformations of ordinary differential operators W. All egretto

Finiteness criteria for the negative spectrum and nonoscillation theory for a class of higher order elliptic operators

9

F. V. Atkinson

A class of limit-point criteria

13

M. F. Ba rns 1ey

Bounds for the linearly perturbed eigenvalue problem

37

M. F. Barnsley, J. V. Herod, D. L. Mosher and G. B. Passty Analysis of Boltzmann equations in Hilbert space by means of a non-linear eigenvalue property

45

John Baxl ey Christer Bennewitz Richard C. Brown

Robert Ca rro 11 J. M. Combes and R. Weder

Constantin Corduneanu A. Devinatz and P. Rejto W. N. Everitt M. Faierman

J. Fleckinger

Some partial differential operators with discrete spectra

53

Spectral theory for hermitean differential systems

61

Wirtinger inequalities, Dirichlet functional inequalities, and the spectral theory of linear operators and relations

69

A survey of some recent results in transmutation

81

Spectral theory and unbounded obstacle scattering

93

Almost periodic solutions for infinite delay systems

99

A Schrodinger operator with an oscillating potential

107

On certain regular ordinary differential expressions and related operators

115

An eigenfunction expansion associated with a two-parameter system of differential equations

169

Distribution of eigenvalues of operators of Schrodinger type

173

ix

x

CONTENTS

The local asymptotics of continuum eigenfunction expansions

181

Some open problems on asymptotics of m-coefficients

189

Singular linear ordinary differential equations with non-zero second auxiliary polynomial

193

R. Kent Goodrich and Karl Gustafson Higher dimensional spectral factorization with applications to digital filtering

199

J. R. Graef and P. W. Spikes The limit point-limit circle problem for nonlinear equations

207

Stephen Fulling Charles T. Fulton Richard C. Gilbert

1som H. Herron

A model problem for the linear stability of nearly parallel flows

Don B. Hinton and K. Shaw Titchmarsh-Weyl theory for Hamiltonian systems

211

219

Christopher Hunter

Two parametric eigenvalue problems of differential equations

233

Arne Jensen

Schrodinger operators in the low energy 1imit: some recent results in L2(R4)

243

Hans G. Kaper

Long-time behaviour of a nuclear reactor

247

Tosio Kato

Remarks on the selfadjointness and related problems for differential operators

253

R. M. Kauffman

A Weyl theory for a class of elliptic boundary value problems on a half-space

267

Ian W. Knowles and O. Race

On the correctness of boundary conditions for certain linear differential operators

279

S. J. Lee

Index and nonhomogeneous conditions for linear manifolds

289

Howard A. Levine

On the positive spectrum of Schrodinger operators with long range potentials

295

Roger T. Lewi s

The spectra of some singular elliptic operators of second order

303

Peter McCoy

Recapturing solutions of an elliptic partial differential equation

319

Joyce McLaughlin

Fourth order inverse eigenvalue problems

327

Angelo B. Mingarelli

Sturm theory in n-space

337

Branko Najman

Selfadjointness of matrix operators

343

A. G. Ramm

Spectral properties of some nonselfadjoint operators and some applications

349

CONTENTS

xi

Thomas T. Read

Dirichlet solutions of fourth order differential equations

355

Martin Schechter

Spectral and scattering theory for propagative systems

361

B. Simon

Spectral analysis of multiparticle Schrodinger operators. Schrodinger operators with almost periodic potentials

369

Estimates for eigenvalues of the Laplacian on compact Riemannian manifolds

371

Phil ip Wal ker

The square-integrable span of locally square integrable functions

375

Stephen D. Wray

On a conditionally convergent Dirichlet integral associated with a differential expression

379

Udo Simon

LECTURES NOT APPEARING IN PROCEEDINGS H. E. Benzinger

Rayleigh-Schrodinger perturbation of semi-groups

C. Bill igheimer

Spectral propertiei of differential operators in the complex plane in B -algebras

P. J. Browne

Eigencurve asymptotics for two parameter eigenvalue problems

H. L. Cycon

On the form sum and the Friedrichs extension of Schrodinger operators with singular potentials

P. Deift

New results in inverse theory

E.

Harrell

H. Kalf

Very small spectral properties of Schrodinger operators

J. Neuberger

On the non-existence of eigenvalues of Dirac operators Operators on L2 (I) ~ Cm Calculation of eigenvalues for -~ + V on a region in R3

S. Ranki n

Generation and representation of cosine families

B. Textorius

Generalized resolvents and resolvent matrices of canonical differential relations in Hilbert space

R. R. D. Kemp

xii

ADDRESS LIST OF CONTRIBUTORS C. D. Ahlbrandt W. All egretto F. V. Atkinson M. F. Barnsley

John Baxl ey Christer Bennewitz C. Bi11igheimer Richard C. Brown Robert Carroll Constantin Corduneanu Percy Deift Allen Devinatz W. N. Everitt M.

Faierman

J. Fleckinger Stephen Fu 11 i ng Charles T. Fulton Richard C. Gilbert R. Kent Goodrich

Department of Mathematics, University of Missouri, Columbia, Missouri 65211 Department of Mathematics, University of Alberta, Edmonton, CANADA T6G 2Gl Department of Mathematics, University of Toronto, Toronto, CANADA M5S lAl School of Mathematics, Georgia Institute of Technolog~ Atlanta, Georgia 30332 Department of Mathematics, Wake Forest University, Winston Salem, North Carolina 27109 Department of Mathematics, University of Uppsala, Uppsala, SWEDEN Department of ~·1athematics, McMaster University, Hamilton, Ontario, CANADA L8S 4Kl Department of Mathematics, University of Alabama (Tuscaloosa), University, Alabama 35486 Department of Mathematics, University of Illinois, Urbana, Illinois 61801 Department of Mathematics, University of Texas at Arlington, Arlington, Texas 76019 Courant Institute, New York University, 251 Mercer St., New York, New York 10012 Department of Mathematics, Northwestern University, Evanston, Illinois 60091 Department of Mathematics, The University of Dundee, Dundee, SCOTLAND, UNITED KINGDOM DDl 4HN Department of Mathematics, University of the Witwatersrand, Johannesburg, 2001 SOUTH AFRICA Universite Paul Sabatier, 118, Route de Narbonne, 118 31062 Toulouse CEDEX FRANCE Department of Mathematics, Texas A & M University, College Station, Texas 77843 Mathematics Department, Penn State University, University Park, Pennsylvania 16802 Department of Mathematics, California State University, Fullerton, Fullerton, California 92634 Department of r~athematics, University of Colorado, Boulder, Colorado 80309 xiii

xiv

Karl Gustafson James V. Herod Isom H. Herron Don B. Hinton Christopher Hunter Arne Jensen Hans G. Kaper Tosio Kato R. M. Kauffman Ian W. Knowles Luis Kramarz S. J. Lee Howard A. Levine Roger T. Lewi s Peter McCoy Joyce McLaughlin Angelo Mingarelli David Mosher Branko Najman Gregory B. Passty A. G. Ramm Thomas T. Read Martin Schechter Ken Shaw B. Simon

LIST OF CONTRIB UTORS

Department of Mathematics, University of Colorado, Boulder, Colorado 80309 School of Mathematics, Georgia Institute of Technolog~ Atlanta, Georgia 30332 Department of Mathematics, Howard University, Washington, D. C. 20059 Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37916 Department of Mathematics, Florida State University, Tallahassee, Florida 32306 Department of Mathematics, University of Kentucky, Lexington, Kentucky 40506 Argonne National Laboratory, Argonne, Illinois 60439 Department of Mathematics, University of California at Berkeley, Berkeley, California 94720 Department of Mathematics, Western Washington University, Bellingham, Washington 98225 Department of Mathematics, University of Alabama in Birmingham, Birmingham, Alabama 35294 r~athematics Department, Emory University, Atlanta, Georgia 30322 Department of Mathematics, Pan American University, Edinburg, Texas 78539 Department of Mathematics, Iowa State University Ames, Iowa 50010 Department of Mathematics, University of Alabama in Birmingham, Birmingham, Alabama 35294 United States Naval Academy, Annapolis, Maryland 21402 Department of Mathematics, Rensselaer Polytechnic Institute, Troy, New York 12181 Department of Mathematics, University of Ottawa, Ottawa, Ontario, CANADA K1N 9B4 School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332 Department of Mathematics, University of California, Berkeley, California 94720 School of Mathematics, Georgia Institute of TechnolQgY, Atlanta, Georgia 30332 -Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109 Department of Mathematics, Western Washington University, Bellingham, Washington 98225 Division of Natural Sciences and Mathematics, Yeshiva University, 2495 Amsterdam Avenue, New York, NY 10033 Department of Mathematics, V. P. I., Blacksburg, Vi rgi ni a 24061 Department of Mathematics, California Institute of Technology, Pasadena, California 91125

LIST OF CONTRIB UTORS

Udo Simon Paul W. Spikes Philip Walker Ri ca rdo vJeder Stephen D. Wray

xv

Technische Universitat Berlin, l-Berlin 12, FRG, WEST GERMANY Department of Mathematics, Mississippi State University, Mississippi State, Mississippi 39762 Department of Mathematics, University of Houston, Houston, Texas 77004 Instituto de Investigaciones en Matematicas Aplicadas y en Sistemas, Universidag Nacional Autonoma de Mexico, Apartado Postal 20-726, MEXICO 20, D. F. Department of Mathematics and Computer Science, Mount Allison University, Sackville, New Brunswick, CANADA EOA 3CO

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Spectral Theory of Differential Operators I. IN. Knowles and R. T. Lewis leds.) © North·Holland Publishing Company, 1981

TRANSFORMATIONS OF ORDINARY DIFFERENTIAL OPERATORS Calvin D. Ahlbrandt

Don B. Hinton

Univ. of Missouri Columbia, MO 65211 U.S.A.

Univ. of Tennessee Knoxville, TN 37916 U.S.A.

Kummer-Liouville coordinate changes order vector differential operators form. This study is preliminary to forms and transformation theory for differential operators.

1.

Roger T. Lewis* Univ. of Ala. in Birmingham Birmingham, AL 35294 U.S.A.

are presented for fourth of the formally self-adjoint the development of canonical linear fourth order partial

INTRODUCTION

This is part of an ongoing investigation of variable change methods for differential operators. The impetus for the general study was a desire to unify results in spectral and oscillation theories for operators having a singularity at 0 and operators having a singularity at The transformation theory for scalar ordinary operators of even order was developed for the real case in [2]. More general results for the second order, including certain partial differential operators, were presented in [3]. An extension of the "Kelvin transformation" to powers of the Laplacian was presented in [4] and a discussion of various equivalences of operators was given in [1]. The present fourth order vector discussion illustrates the theory for higher order vector ordinary differential operators and builds notation for the fourth order partial case. The transformation theory for the odd order cases is obtained as a corollary to the even order cases. 2.

THE SECOND ORDER CASE

consider the second order scalar Jacobi-Reid [8] canonical form L[y]

=

-(r(x)y' + q(x)y)' + (q(x)y' + p(x)y).

(2.1)

Suppose that p and r are real valued and q is complex valued on a real interval x. The special case where q is real valued arises in the Calculus of Variations [6]. I f p, q, and r are continuous and r never vanishes on x, then the "off-diagonal" terms may be removed by a variable change to produce a two term operator [3,7]

=

L[y]

(2.2)

-(r(x)y')' + p(x)y.

However, the form given in (2.1) has several advantages over the form (2.2). First, the form of (2.1) is preserved under Kummer-Liouville coordinate changes [3, TH. 2.2] y(x)

=

~(x)z(t),

t

=

f(x),

~

and

f'

nonvanishing,

(2.3)

with ~ complex valued. (The form of (2.2) is preserved in case ~ is real valued, but not necessarily if ~ is complex valued.) Second, the generalization of (2.1) to the vector case [8] L[y]

=

-(R(x)y' + Q(x)y)' + (Q*(x)y' + P(x)y,

(2.4)

(here P, Q, and Rare n x n complex matrix valued with P and R hermitian on X), includes a useful first order case. Indeed, the special case of (2.4)

2

G.D. AHLBRANDT et al.

with P hermitian, R - 0, and the "Atkinson form" [5]

Q

L[y]

a constant skew hermitian matrix reduces to

= Jy '

+ P(x)y

(2.5)

for J defined as 2Q*. Third, the form of (2.4) is needed for general KummerLiouville transformations

= H(x)z(t),

y(x)

=

t

f(x),

H nonsingular,

f'

~

of (2.4) even if all the involved matrices have real entries. of (2.4) under (2.6) is of the form [3] LO[Z]

0,

(2.6) The image operator

-(ROZ' + QOz)' + (QOz' + Poz)

=

with the coefficient matrices being functions of t conditions on P, Q, R, H, and f the operators the identi ty

on L

(2.7)

T = f (X). Under certain and LO ·are related by

{(l/!f'!)H*L[y]}(x) = LO[Z](t) for

(x,y)

and

(t,z)

(2.8)

related by (2.6).

A generalization of the "Atkinson form" to partial differential operators was included in [3]. 3.

A CANONICAL FORM FOR FOURTH ORDER ORDINARY OPERATORS

If the vector operator L[y]

(R (x)y")

",

R* (x)

= R (x) ,

(3.1)

is subjected to a variable change of the form, y(x) = H(x)z(t), with t = f(x), f is real valued of class C4(X), f' (x) never vanishes, H is n x n complex valued of ~ C4 (X), and H is nonsing;D:'ar ~ X,

(3.2)

then a natural canonical form for fourth order formally symmetric operators evolves. A sufficiently general form for variable change purposes is 2

L[y]

=

l:

.

(-l)~

i=O

~

A(i,j)y(j)}(i)

(3.3)

j=O

where each coefficient A(i,j) (x) on a real interval X such that

is an

A*(i,j) = A(j,i),

i,j

n x n =

complex matrix valued function

0,1,2.

(3.4)

If the indices of summation in (3.3) are allowed to run to m, a rather general 2mth order formally symmetric quasidifferential operator is obtained. If the coefficient A(m,m) is zero, then the operator is of odd order and the transformation theory for those cases can be obtained as a special case of the theory for the even order case. The discussion will be restricted to the fourth order case since it is typical of the higher order cases. 4.

KUMMER-LIOUVILLE TRANSFORMATIONS

In order to fix the setting, let efficients in (3.3). (H)

Ci(X), ~,J = 0,1,2, and the matrix i,j = 0,1,2, is hermitlan-.- - - 4 is in the domain of L if Y is of class C (X). Set

y

Suppo-6e hypothrv..,u, when app,Ue.d:to L[y]

THEOREM

L [z]

o

assume the following hypothesis on the co-

A(i,j) is of class A(x) = (A(i-;j)(Xj),

Suppose that (3.2)

US

=

T

=

f(X).

(H) ho.e.dJ... The KwnmeJt-UouvLUe vaJUable c.hange 06 (3.3) geneJta.trv.. an opeJta.tOIt LO 06 the 601tm

{(P z" + Q Zl)" 22

(Q*z" + P z' + Q z)' + (Q*Z' + P z)} 211 10

(4.1)

3

TRANSFORMA nONS OF ORDINAR Y DIFFERENTIAL OPERA TORS

.6Uch ;tW ;the identity (2.8) hold6. FuM:heJunolLe, ;the P. a.Ytd Ci(T) wdh P. heJrm.(;tianand Q. 6k.ewheJun);t[an. Aub 1

Q i

Me.

06 c..R.a.M

1

3 P (t) = {!f'1 H*A(2,2)H}(X) 2

(4.2)

and poet) = {(l/lf'l) (1/2) (H*L[H] + (L[H])*H)}(x)

(4.3)

An algorithm which yields the remaining coefficients is provided by our constructive proof. In general, the remaining coefficients are quite complicated. However, we now list several "elementary" examples.

SUppo.H. A(i,j) = 0, - a for B(~,~) > K(~,~)

which

for all

~

€

Coo(P). o

This is an adaptation of the defini-

tion introduced by Glazman, [5]. A summary (up to 1973) of conditions for B or 9. to be nonoscillatory or oscillatory can be found in the books of Swanson, [IS], and Kreith, [7]. For more recent criteria and extensions to more general cases and to related problems, we refer to the results of MUller-Pfeiffer, [10]; Kusano and Yoshida, [8]; Hinton and Lewis, [6], and the references mentioned therein. THE CASE

m= 1

We consider first the case where £ is the second order expression: 9.~ = -/::,~ - q~. Suppose that q is regular "in bands". That is: there are smooth surfaces (i)

q

€

{Rk}~=O' tending to

Cl[Mk n G] n Loo(N

k

n G)

00, such that:

where

are neighbourhoods of 9

10

W, ALLEGR1:'TTO

Rk n G respectively; CE)

q

Ciii)

q

(iv)

L'" Q,oc in a neighbourhood of dG; is of class Ln/2 in any bounded subdomain of

is of class +

the domain bounded by

R , R (p

>

q)

and

G

G;

can be expressed as

p q u Z G with \l (Z ) and G a domain such that if pq pq pq = pq / then q E Lr 2 CT ) with r = rCT) > n.

°

T

cc G

pq

The above assumptions are a particular case of the ones introduced in [1], [2]. The following theorem is a consequence of the results established in [2]. Theorem 1. is finite.

Let

q

be regular "in bands".

Then

B is nonoscillatory iff a _ (1)

Related results have been established by Piepenbrink, [11]. and Moss and Peipenbrink, [9]. We remark that Theorem 1 remains valid if in the expression for Q, we substitute - I O. (a .. D.q,) for -6q" as long as the a., are 1

1J J

1)

reasonably regular (see [2]). We assume in the sequel, without further mention, that at least the above conditions hold on q. THE CASE

m

>

1

Serious difficulties appear to arise when an attempt is made to extend the arguments of Theorem 1 to the case m > 1. Indeed, it does not appear known in this case if the finiteness of o_CL) follows from the nonoscillation of B. We show, however, that if B is nonoscillatory by iteration of second order arguments then o_CL) is finite. Let Q denote the subset Cwo'" .,wm_l ) belongs to

of [C'" (G)] m with positive components such that Q iff Wa _ 1 and the forms:

BkCq"q,) =

n

J

W

G are nonoscillatory for

k

k = 0, ... ,m-2.

I

i=l

Note that

can always be chosen near infinity of type constants A,a,S. Theorem 2.

Let

(D q,)2 - w + q,2 i k l

Q is not empty, since the wk

Alxla(log Ixl)S

and suppose that the form

f G

B'

with suitable given by:

w _ ~ (Oi.)2 _ q.2 m l 1

is also nonoscillatory. Then there exists a finite number of linear functionals p '" such that l' f q, E P~ {Null space f } then BC.,q,) > o. (fi}i=l on Co(G) i Proof.

We express

B in the form: B(q,) =

I

I I B. 2 m j=2I a i l=l J-

(oaj+ ... +amq,)!

+ B' C.)

m~i.::j

where

a. )

is a nultinomial and

B(

- M - woq,

o

Wi (woD i ¢) - q¢

o

If we follow a procedure introduced for a different 2 w = div P - IpI > 0 and

problem by Protter, [13], we find that we may choose: -1

2

q:. div s - Wo lsi, where s.

P

=

(PI,···,P n ), S

=

0

(SI"",Sn)

and

Pi'

Let us further assume that G is an excerior domain. One choice 1 of P gives W (n_2)2 4- l lx l-2 (near 00). If q is specialized to be of o type alxl- 4 near then, by this method, we obtain a = (n_2)2(n_4)2 4 -2. In this special case the "optimal" a is known to be n 2 (n_4)2 4-2, for n > 4, and E

COO (G).

was obtained, [3], by nonoscillation theory, separation of variables, and estimates which depend strongly on the nature of the specific problem considered. It is interesting to note that the above "optimal" value of a is also exactly where B changes from oscillation to nonoscillation. In the above case, our method gives a worse result then what was previously known. To give a simple example of a result which does not seem obtainable by other methods we state: Example 2. Let n the cone x3 = alxl if

Ix I

lead to


0, 3G is described by (a near 1) if Ixl > R, while 3G is essentially arbitrary

Then the above arguments together with some related estimates, [4], being finite if qlxl 4 :. (9_a)2(l_a)-2 4 -2 near

o_(l)

D ¢, D ¢ as independent i j functions and appears to "change" the side boundary conditions (heuristically, from u = dU/dn = 0 to u = ~u = 0 if m = 2). It would be desirable to remove these shortcomings, but it is not clear how this can be accomplished in general.

We conclude by remarking that the method treats

Finally, we note that the localization procedures which we have introduced imply that operators with a singularity at a finite point of the boundary and/or multiple singularities can be handled in the same way, at least formally. While we do not pursue this point, we note that it may be very difficult to obtain explicit nonoscillation criteria for the above cases unless the geometry of the problem is simple near the singular set.

W.ALLECRETTO

12

REFERENCES [1]

Allegretto, W., Positive solutions and spectral properties of second order elliptic operators, Pacific J. Math., to appear.

[2]

Allegretto, W., Positive solutions of elliptic operators in unbounded domains, J. Math. Anal. Appl., to appear.

13]

Allegretto, W., Finiteness of lower spectra of a class of higher order elliptic operators, Pacific J. Math. 83 (1979) 303-309.

[4]

Allegretto, W., Nonoscillation criteria for elliptic equations in conical domains, Proc. Amer. Math. Soc. 63 (1977) 245-250.

[5]

Glazman, I.M., Direct methods of qualitative spectral analysis of singular differential operators, Israel Program for Scientific Translations (Davey and Co., New York 1965).

[6]

Hinton, D. and Lewis, R., Oscillation theory for generalized second-order differential equations, Rocky Mountain J. Math. 10 (1980) 751-766.

[7]

Kreith, K., Oscillation theory (Lecture Notes in Mathematics, Vol. 324, Springer-Verlag, Berlin 1973).

[8]

Kusano, T. and Yoshida, N., Nonlinear oscillation criteria for singular elliptic differential operators, Funkcial. Ekvac. 23 (1980) 135-142.

[9]

Moss, W. and Piepenbrink, J., Positive solutions of elliptic equations, Pacific J. Math. 75 (1978) 219-226.

[10]

Muller-Pfeiffer, E., Ein oszillationssatz fur elliptische differential gleichungen hoherer ardnung, Math. Nachr. 97 (1980) 197-202.

[11]

Piepenbrink, J., A conjecture of Glazman, J. Differential Equations 24 (1977) 173-177.

[12]

Reed, M. and Simon, B., Analysis of operators (Academic Press, New York, 1978).

[13]

Protter, M.H., Lower bounds for the first eigenvalue of elliptic equations, Annals of Math. 71 (1960) 423-444.

[14]

Schechter, M., Spectra of partial differential operators (North Holland Amsterdam, 1971).

[15]

Swanson, C.A., Comparison and oscillation theory of linear differential equations (Academic Press, New York, 1968).

Spectral Theory of Differential Operators I.W. Knowles and R. T. Lewis (eds.) © North-Holland Publishing Company, 1981

A CLASS OF LIMIT-POINT CRITERIA F. V. Atkinson University of Toronto Limit-point criteria for second-order differential operators, and limit-n criteria for 2n-th order operators, generally impose a positivity condition on the coefficient of the highest derivative, and bound other coefficients relative to it. This paper deals with criteria which focus attention on the coefficient of the independent variable, and which make no requirements of positivity or reality.

1. INTRODUCTION. We are concerned here with the classification problem for differential equations of the form - (p(x)y')' + q(x)y = AY, a < x < 00 (1.1) and, to a lesser extent, with certain variations and generalizations, such as matrix equations, higher-order equations and difference equations. For (1.1), this problem goes back to the fundamental papers of H. Weyl (30, 31), who found that just two cases were to be distinguished. If one denotes by d(A) the dimension of the space of solutions of (1.1) which are of integrable square on (a, 00), these cases are: (i) the limit-circle case,

in which

(ii) the limit-point case, in which Here

=2

d( A) d( A)


0, a < x < The terms "limit-circle", "limit-point" can then be validated in terms of. the behaviour of certain circles in the complex plane; however the classification is sound without any reality or positivity hypotheses on p , q .

Weyl also proved, among much else that in the real case d (A) > I if 1m A I O. These discoveries provided the prime examples of the theory of deficiency indices of linear operators and of their extensions, and the main impetus for the extensive development of this theory in the context of differential operators in recent years. Generalization of the theory beyond Weyl's hypotheses, that p is real, positive and continuous, and q real and continuous, may be seen as a staged process. Customary assumptions are now that p(x)

>

O.

P

-1

, q

(1. J)

E

i.e. are in L(a, b) for every b E (a,oo). This allows p(x) to vanish or become infinite for individual x-values, giving rise to situations which fall outside the scope of the usual existence 13

F. V. ATKINSON

14

and uniqueness theorems. One may avoid any difficulties in this connection by interpreting (py') as a "quasi-derivative" or, better, by going from (1.1) to a first-order system. This is to be accomplished in a known manner by setting py' = z, P -1 = r , (1 .4) so that (1.1) can be replaced by y' = rz,

z' = (q - A ) y.

Here the coefficients on the right are in L (a,oo), and the solutions y, z will be continuous, indtga locally absolutely continuous functions; z will be well defined, even at points where p, y' fail to be so. A possible generalization which emerges at this point is (as suggested by Everitt) to permit p(x) to change sign, remaining real-valued, along with q(x) . Of course, we must make the restriction that l/p(x) should remain integrable at this point of change of sign. Here again the systems formulation (1.4-5) seems to allow a more natural formulation, in that r(x) can not only change sign, but also can vanish over intervals. As may be see'1 from (1), Chapter 8), much of the standard theory, including the nesting-circle phenomenon, carries over to this case; the assumption made there that r(x) > 0 was needed mainly for the semi-boundedness of the spectrum. It would seem that the detailed analysis of this case presents a considerable challenge. In another direction, one dispenses with the hypothesis that q is real-valued, and perhaps also for p ,so that there is no question of formal self-adjointness, but still retains a positivity hypothesis for p, either for p itself or for its real part if p is complex. This has been extensively investigated recently by Knowles and Race (20, 25) and others. The emphasis of the present paper will be on criteria for the presence of a non-integrable-square solution, when no hypotheses are made concerning the reality or positivity of the coefficients. The criteria will involve mainly bounds, pointwise or integral, placed on q rather than on p . These criteria will appear as special cases of general, necessary and sufficient conditions for the existence of such a solution. One approach, though a restrictive one, to the determination of d( A) is that of asymptotic integration. Subject to various restrictions, involving among other things differentiability conditions on p and q , one can approximate to the solutions, and so test their square-integrability directly.; slightly less restrictively, one may sometimes be able to test square-integrabil-ity by investigating the behaviour of suitable energy-type, or Lyapunov (or Kupcov) functions. In the reverse direction, it can be said that the proofs of weaker and more general limit-point criteria can be adapted to yield quantitative information on, so to speak, the non-square-integrability of solutions. Results of this nature were given in (4), and we shall include some here. 2. A SELECTION OF CRITERIA FOR THE CASE

P

>

0, OR

Re p

>

O.

For the sake of comparison we review some of these briefly; we start with two rather classical sufficient criteria for the limitpoint case, when p is real and positive and q real. The first is simpler, and is among the original results of Weyl (Jl). I. q(x) is bounded below on (a,oo).

A CLASS OF LIMIT-POINT CRITHRIA

15

The proof is immediate from the observation that if in (1.1) - q(X) < 0 for large X, then there is a solution which is ultimately positive and increasing, and so not s'1uare-integrable. \

Partially overlapping with this, but allowing q to become large and negative, is the criterion of Levinson (4, 21). II. There should exist a positive, locally absolutely continuous function W such that q > - W , and 2 3 1 sup pW' W< 00, J (pW)-2"dx = (2.1-2) 00

The criterion II includes I by taking not being in L{a,oo).

W

=

1

1 , subject to

p 2"

These criteria are of the global type, and have the feature that q is bounded on one side, as is of course p. In the case of II we have the hypothesis of the existence of an auxiliary function W , linked with p and q by inequalities. Subsequent developments involve restricting p and q on a sequence of intervals only, or allowing them to take complex values, or the introduction of a greater number of auxiliary functions, or again the use of integral rather than pointwise bounds on p and q. We illustrate these points in the following examples. III. The Levinson criterion II retains its validity if q is allowed to be complex, with q ~ - W being replaced by Re q ~ - W, other conditions remaining unchanged (3). The imaginary part of q plays no part in the criterion, and is arbitrary. With q complex, we can no longer speak strictly of the "limit-point case", but have rather a sufficient condition for the "J-selfadjointness" of certain operators. See (25) for more details. Going back to the real case, we have the criterion IV. We take p = 1, and assume that q(x) has a fixed lower bound on a sequence of intervals of fixed positive length, with disjoint interiors. This forms a very special case of results of Hartman, since developed further by Eastham and others (see (5»; one can also adapt criterion III so as to cover this condition (see (3). The criterion shows that the limit-point case can remain quite unaffected if p, q are left arbitrary on large parts of the axis. Moving on to the case that p may also be complex, we cite the following interval-type criterion which, though not quite the most general available, is reasonably simple, involves no auxiliary functions, but rather a choice of intervals and parameters. See (2). V. On a sequence of non-overlapping intervals (a, b ), let m m (i) Re pet) > Mm > 0, ip(t)i ~ KMm' am < t < b m (2·3) B

< a < S < b , ( 2.4) (ii) (bm - am) f Re q dt > - Km m ill a and ( iii) I (bm - am) 2/Mm = (2.5) m In this result, pet) must lie in a certain fixed sector in the right half-plane, Re q(t) satisfies a one-sided integral bound, and a sum (2.5) must be infinite (just as the integral (2.2) must be infinite). These features, or slight modifications of them, can be recognised in almost all existing limit-point criteria. 00 •

Interval-type conditions can be brought within the scope of global conditions by employing an auxiliary function which vanishes

16

F.V. ATKINSON

outside the intervals. For further developments we cite the papers of Knowles and Race (20). Read (28) and Frentzen (9). together with the survey article (8) and monograph (18). We pass now to the distinct type of limit-point criterion which forms our main concern here. 3. INTEGRAL-TYPE LIMIT-POINT CRITERIA ON q. The arguments in the sequel are largely suggested by t~e remarkable observation that (1.1) has a solution not in L (a. oo ) if

(3.1) L2 (a,oo). Here p. q may be complex-valued. and p is quite arbitrary. subject to our general conditions P -1 • q E Lloc(a,oo). (3.2 ) The above assertion. in the case p = 1 and q real. is due to Hartman (12). in whose paper it appears as a special case of the criterion q E LS(a,oo), for some s :: 1; this in turn is linked with the non-oscillatory character of (1.1) in this case when A < O. The criterion (3. i) for limit-point. with p = 1 and q real. is sO:,letimes attributed to Putnam (24). whose contribution was. however. to elucidate the nature of the spectrum in this case; I am indebted to Professor H. Kalf for clarification on this matter. A short proof of (3.1) as a limit-point criterion. with arbitrary p • is given in (18). It does not appear that there is any limit-point criterion which restricts p only. leaving q arbitrary. There have been a number of developments regarding the criterion (3.1). It has been shown by Zettl (33) that (3.1) ensures the existence of a non-integrable-square solution in the case of a class of higher-order equations (with q being still the coefficient of the dependent variable). For a slightly narrower class of higher-order eauations. Hinton (15) has given the more general criterion JT I 2 I q (tl dt = OCT), as T .... (3.3) q

E

00,

o

as sufficient for a certain bound on the dimension of the set of of L -solutions; he has also extended the result to solutions in other Lebesgue classes. In this section we go back to the second-order case - (py')' + qy = O. a 2. x < 3 . 4) with finite a • and obtain a criterion which is slightly more general than (3.3). and of course than (3.1). We also obtain a quantitative estimate of the "non-square-integrability". Theorem 1. Let -1 2 P E Lloc(a,oo), q E L loc(a,oo). (3·5) 00 ,

are solutions of (3.4) satisfying v(pu') - u(pv') = 1. we have, for some C • and writing 2 2 w = lul + Iv1 , x 2 x t 2 J w dt :: {C + 2 J (C + J Iql ds)-l dt}Yz - C . a a a In particular, (3.4) has a solution not in L 2 (a,oo) Then. if

(

u. v

(3·6) (3.7) (3.8) if

17

A CLASS OF LIMIT-POINT CRITERIA

x

2 Igl dt}-l ~ a In particular, the conclusion holds if {l +

T

J

J

2 Igl dt

=

L(a,oo).

(3.10 )

OtT log T), as T ~ 00,

a

or again if

q(x)

=

1

0(log2x ).

(J.ll)

Proof of Theorem 1. This consists of a slight development of the method used to Justify the criterion (J.l). From (J.4), (J.6) we deduce that x x 1 = v(x) {(pu') (a) + J gu dt} - u(x) {(pv') (a) + J gv dt} . (3.12) a a Hence, if 2 2 k (J.1J) C = {I (pu' ) (a) I + \ (pv' ) (a) \ } 2 , we have from (3.12) that X J.:. k (J.14) 1 < Cw 2 (x) +W2(X) J I g (t) IwYz (t) d t a x x k (3.15) < w 2 (x){C + (J Igl2 dt J w dt)Yz}. a a Squaring, we deduce that x x 2 1 :: w(x) {C + J w dtj{C + (3.16) J Igl dt}. a a Dividing by the last factor and integrating we obtain t

x

J

{C +

J

Igl2 ds}-l dt

a a from which (J.8) follows easily.

x


0, and let f, g be positive-valued functions on ~), with f locally integrable and g continuously differentlable and non-decreasing, such that x

A ::

f(x) + f(x)

Then

x

J

2

f

2

(t) dt ~ A

J f(t)g(t)dt, a x

J

(1 + 2A

t

J

a

x

=

f(x)g(x) {l +

J a

(4.1)


A I (g(t))- (g(t))(A+h(t)){l + 2 I (A+h)g ds}-l dt. a a a Here we integrate by parts, and get x 2 1 t x I f dt > lzA[(g(t))- log(l + 2 I (A + h)g ds)la + a a + lzA

I

x

g'(t)(g(t))-2 log(l + 2

J

t

(A + h)g dS) dt.

a

a

Since g' > 0, h ~ 0, the right-hand side is not increased if we replace h-by O. Doing this, and reversing the integration by parts, we obtain the required result (4.2). We now obtain a pointwise analogue of Theorem 1. Theorem 2. Let p, q satisfy (J.2), and let Iq(x) I ::. g(x) , a::. x < 00, (4.J) where g(x) is positive, non-deceasing and continuously differentiable. Let u, v be as in Theorem 1. Then, for some A E (0,=), x 2 2 2 x t I (lui + Ivl )dt ~ A I (1 + 2A I g(S)ds)-l dt. (4.4) a

a

In particular, if q(x)

=

a

O(log x), as

then (J.4) has a solution not in

x

-> 00

L2(a,00).

The bound (4.4) follows from the application of Lemma 1 to (J.14). It is immediate that if (4.5) holds, we may take g(x) to be, for large x , a multiple of log x , so that the right of (4.4) will become unbounded as x -> Again, we can improve (4.5) by inserting additional factors on the right invo.lving iterated logarithms. The bound (4.5) is, of course, an improvement of (J.ll). 00

5. DISCUSSION OF THEOREMS

1

AND

•

2

We can check the precision of Theorem 2 by means of asymptotic integration. We need the rather standard Lemma 2. Let f, g be positive-valued and continuously twice differentiable on [a,oo), and let g-

1

1

1

({2" g2)'

E BV [a, 00) ,

(

5 . 1)

19

A CLASS OF LIMIT-POINT CRITERIA

i.e. be of bounded variation over the whole semi-axis, and let also lim sup ->- 00 Then a solution

(5.2)

x

y

of ( fy' ), + gy

=

0

(5.3)

x ,

satisfies, for large

( 5.4) The proof follow0 a Kupcov-style argument, using the energy function 2.1. 2 -.1. --1.1. .1. E = Y (fg)2 + (y'£) (fg) 2 + g (f 2 g 2 )'fyy' , for which E'

We omit further details. We apply this to the example ((x 2 1og xly')' + (log x)l+c y and deduce that y(x)

=

_lo.

O(x 2(log x)

=

0,

>

0,

(5.5)

__1.._lo.s

2 2 ), 2 so that y E L (2,00). It follows that in the criterion (4.5) the power of log x cannot be increased, if this is to serve as a sufficient criterion for the limit-point case.

We can also test Theorem 2 in respect of the growth of the integral on the left of (4.4). Thus, if q(x) is bounded, we have a result of the form, for large x , x 2 2 f ( Iu I + Iv I )dt > 0 log x, (5.6) a D > O. This this is a correct order of magnitude may for some be seen in the case of the Euler equation 2

(x y' )' + Y = O. l

(5.7)

In this particular case, Theorem 1 gives (5.6) with log2 x in place of log x ; however it yields this result under the more general assumption (3.3). Whether (3.10) is in some sense best possible is not clear. However it is evident from the case of (5.5) that the power of log x on the right of (3.10) cannot be replaced by any power greater than 2 . 6. SOME VARIATIONS. We first note the adaptation of Theorem 1 to first-order two-dimensional systems of the form (1.5); it is sufficient to take the case A = O. This will permit an application to secondorder dif~erence equations. Theorem 3. Let 2 r E Lloc(a,oo), q E L loc(A,oo), (6.1) and let (3.9) hold. Then the system y' = rz , z' = qy , has a solution for which

y

is not in

(6.2) L 2 (a,oo).

For the proof we take a pair of solutions of (5.2) such that y l z2 - Y2z1 = 1 , so that

F.v. ATKINSON

20

for Some constants c l ' c 2 ' and argue as in Section J. In particular, we can use this result with the roles of interchanged, to get Theorem 4. Let -1 2 P E Lloc(a,co), q E Lloc(a, 00),

r, q (6.4)

and let

(6.5) a Then (J.4) has a solution such that py'

%

L 2 (a , 00) •

(

6.6)

For example, if p is bounded, then there is a solution such that y' is not square-integrable. In the case p = 1 this is a result of Hartman and Wintner (lJ). Still with p bounded, we can conclude from (6.6) that there is a solution such that 1

p2y'

%

L 2 (a,00),

(6.7)

(17)) of (J.4) does not exceed

so that the Dirichlet index (see

1

Next we remark that the argument of Section J can be pursued in other L-spaces; we assume (J.2) and omit detailed proofs. q

(6.8)

E L(a,oo),

then (J.4) has a solution which does not tend to zero. Theorem 6. If, as +

0,

then (J.4) has a solution not in Theorem 7.

Let a, 8

E

q

(1,00)

L(a,

00).

1, and let

satisfy lla + 1/8

E L~oc (a,oo),

(6.10) (6.11)

L(a,oo) •

a Then (J.4) has a solution not in

L 8 (a,co).

Here Theorem 7 is an extension of Theorem 1. For extensions to higher-order equations we refer to the paper of Hinton (15). Illustrating these results, we observe apropos of Theorem 5 that (xl+ -1 = 0(1), 04> -1 f/. L(a, 00'). (9.4-6) Let also p

=

o(

4> ),

p'

= o (cp a -1 ),

q

Then (3.4) has a solution not in

-2 0 (cpa ). L 2 (a,oo).

=

We now use adjoining intervals, with

bm

=

a + m l

' and take

(9.10) a l = a, a m+ l = am + a (am)' (2). We take the as in the proof of Theorem 10 in the paper now need that as in (9.1) and, writing am for a (a ) In

'

The hypotheses (9.4-5) ensure that art) / a(a m ), cp(t) / cp(a ) and their reciprocals are bounded in (a, a +1)' so that tnlr integral in (8.11) is of order cp 2 (aIn)aIn~ Th~s (8.11) will hold if 00 2 L am / cp(a ) 1

In

and, by the above remarks concerning ¢ and a , this is ensured by (9.6). This completes our sketch of the proof. In particular, we can take cp = 1, a (x) = x- l , and conclude that the conditions p ( x) = 0 ( 1) , p' (x) = 0 (x) , q (x) = 0 ( x 2 ) , (9.12) are sufficient to ensure the existence of a solution not of integrable square. The case of real positive p , and possibly complex q, is considered by Hinton (14) as a special case of a result for the 2n-th case. In some later work (see e.g. Frentzen (9)) p may be complex but lies in a sector in the right half-plane, but cannot be arbitrarily small. Since the conditions of Theorem 10 are both necessary and sufficient, it must in principle be possible to obtain from it other types of sufficient criteria, such as those which make onesided restrictions on q or its real part. It would seem that this can be done by Ym in the form vmy, where y is a solution and vm

a factor designed to bring bout (8.2) at, say, the mid-point of

(am' b m)· However the details seem to be repetitive of the known arguments for the standard tests, and will not be taken up here.

25

A CLASS OF LIMIT-POINT CRITHRIA

10. THE CASE OF A FINITE SINGULARITY. If we are considering (3.4) over a fi~ite interval (a, b), and ask whether there is a solution not in L (a, b), we can no longer derive benefit from the arguments of Sections 3 and 4. However Theorems 8 and 10 can still be used, with the obvious modifications. Thus, using Theorem 8 with b in place of ,and taking y (b - x)-t , we have that there is a solution not in L2(a, b) 00

if

p(x)

=

2

O«b - x) ),

p'(x) = O(b - x), q(x) = 0(1),

which can be checked in the case of an Euler equation. From Theorem 10 we can derive interval-type tests of a similar character. 11. THE SECOND-ORDER MATRIX CASE. We extend the above considerations to the equation - (Py')' + Qy = 0.. a < x (11.1 ) < 00 where P, Q are n-by-n matrices of functions, and y is an n-by-l column-matrix of functions. For a general formulation. we assume that P has almost everywhere an inverse R , which is locally Lebesgue integrable, as is Q; these integrability conditions are imposed in fact on the entries in these matrices. Using the quasi-derivative z = Py, we can then if necessary pass from (11.1) to the first-order system y' = Rz, for which a solution

y,

z

z' = Qy ,

(ll.2)

will be locally absolutely continuous.

We denote by I· I any convenient norm for matrices. satisfying the usual requirements. By a superscript (T) we indicate the formal transpose. We have then an almos-s complete extension of the criterion (3.9) Theorem 12. Let

IX

IQI2 dt}-l !Ja Then (11.1) and the transposed equation {l

+

- (y.p). + yQ

L(a,oo).

(ll.3)

= o.

(ll.4)

where y is a row-matrix, cannot both have more than n linearly independent solutions in L2(a,00). Here the 'term L 2 (a,00) is to be interpreted elementwise. As in the case of (11.1), we can pass from a second-order equation to a first-order system z' = yQ , y' = zR. (ll.5) with row-matrices y, z . We suppose if possible that both (11.2) and (11.5) have more than n linearly independent solutions in which y is of integrable square.

If Yl' zl form a solution of (11.2), and of (11.5), we have

Y2' z2

a solution (ll.6)

Here the left provides a non-degenerate bilinear form, with arguments in spaces of complex dimension 2n. Hence, if we have an (n+l)-dimensional space of pairs y, Z ,and likewise of Y?' z2 ' we can choose these so that th~ cotstant in (11.6) is not zero, and is for example 1 . We may thus suppose that the right of (11.6) is 1 , so that

26

F. V. ATKINSON

x

x y 2 Q d t ) y 1 ( x ) - Y2 ( x) ( z 1 ( a) + f Qy 1 d t) . a 2 a Assuming that Yl' YZ E L (a, ro), and making minor modifications in the argument of Sectlon 3. we then get a contradiction with (11.3). 1

=

(z 2 ( a) +

J

In particular, we have the conclusion that (11.1) has at most n linearly independent solutions of integrable square if (11.3) holds, and if P. Q are formally symmetric (i.e. equal to their transposes), or again if the are hermitian symmetric. Systems of somewhat more general form than (11.1), in which P enjoys some positivity property, have been considered by Frentzen (13). 12. EXTENSION TO FIRST-ORDER CANONICAL SYSTEMS. We now extend this type of reasoning to systems of the form Jy' = A(x)y ,

a'::'

x




1.

Remark. In [4J the convexity bound is given for j = 1. There also, a variety of applications to theoretical physics is considered. It seems clear from Figure 1 that much advantage can be gained with j chosen freely.

39

BOUNDS POR THE LL\'L1RLY PliR1URBliD lilGENVALUE PROHLEM

y

Figure 1 The bound A (x) < A (0)+:\ (l)x 1

-

3

3

is saturated for x E [2,3]. y

A 2 (x)

t

o

+

'.

x

(2)

Theorem 2. Let k c {1,2,"',Nj and Ak ~ O. Then real polynomials Pix) and Q(x), of degrees 1 and 2 respectively, can be chosen so that 4 Pix) Ak(X) - Q(x) = 0(x ) and P(O) = 1; moreover A2 (X)

~

Q(x)/P(x)

for all xcI such that A~2)_ ~A~3)x > O.

The bound is best possible

on the basis of the information which it uses. Remark: The function Q(x)/P(x) is the ~2/1J Pade approximant (P.A.) to Ak(x). Also, the tangent A~O)+ XA~l used in Theorem 1 is in fact the [l/OJ P.A. to Ak(x). So far, roughly speaking, we have asserted that any [l/OJ P.A. bounds Al(x) whilst any [2/1J P.A. bounds A2(x) A highly conditional assertion about the relationship between any [M/(M-l) J P.A. and AM(X) for t1 c {1,2,···,N} may be valid. Remark. For the eigenvalues of the linearly perturbed operator

~

+ x 2 + yx4} in L2, dx2 where y is the perturbation parameter, Simon et al. [2,3J have established Stieltjes characterizations which enable bounds to be imposed using P.A. 'so {_

Wilson et al [7J have demonstrated an invariance property which suggests that, from among the various rational approximants which might be used for linearly perturbed eigenvalues, the [M/(M-l) JP~}s are the most appropriate ones.

40

M.F. BARNSLEY

Example.

~) ~

Suppose we have the information ,(0)= 0

Al

A (1)= 1 1 '

'

A(2)= -1 and A(3)= -3. 1 ' 1

Then Theorem 2 yields the upper-bound marked a in Figure 2. formation

S)

, (0) _

112

-

2

, (1)

,

112

=

-1

'

1.(2)= 1, and 2

provides the bound labelled S. simultaneously valid is where

, (3)

=

The in-

3

"2

'

One system for which u) and S) are

A+xB = (x x for which A (x) is the curve marked I in the figure. 2

Figure 2 a and S are [2/1J PA upper bounds for I which denotes a 1.2 (x) .

4

2

-2

-4

x

(2)

Theorem 3 Let Al ~ O. Then real polynomials P(x) and Q(x) , of degrees 1 and 2 respectively, can be chosen so that 2

Al (x) + P (x)

4

Al (x) + Q (x) = 0 (x ),

and 2

A (x) + P(x) 2 Moreover

A (x) + Q(x) 2 for all x

= E

O(x).

I,

where A (x) denotes the lowest root of the equation 2 A (x) + P(x)

A(X) + Q(x) = 0;

and this bound is best possible on the basis of the information which it uses.

41

BOUNDS FOR THE LINEA RL Y PDR TURBDD mCDNVAL UD PROBLDM

OUTLINE PROOFS OF THE THEOREMS We use the notation for the inner product in h. We write for an eigenvector such that

~.

(x)

J

(1) = A.(x)~.(x), and q.(x),l/J.(x» = 1 J J J J J for all x in some neighborhood of O. We let N denote a complex neighborhood of 0 such that, for each j, A.(X) and ~. (x) are regular J and bounded for all x EN. J k k 2k+l L I A(P) P O( 2k+2) ( 2) n=O n. J n=O J p=O PT j x + x , for x E N, (A+xB)~.(x)

-l,

and k
' ' ,2.

'" Lemma 2.1. If x and y are in;', 2 and {zn}n=O is given by zn 2 1 n+1 Lp=O xn_pYp' then Z is in;', and I z I O> = l/(l+b),

<x,1>I> = 0, and

IX-1>O/(l+b) I < 4(1+3b)/3(I+b).

one [unction u:

[0,00)

Then, there is only

~

.(2 such that u is a solution o[ (TW) and t Moreover, for that solution, lu(t)-¢o/(l+be ) I 4- 0 as t

ufO) = x. increases.

Indication of proof.

Let b and x be as supposed and m be a positive

number such that m < 4(1+3b)/3(I+b). set described by C

Let C be the closed, convex

{z: = 0, I> = 0, and

J(t) be the function given by J(t)z = A(z,z) Then J(t): C

~

Izl ,; m}. Let t + ZA(¢o,Z)/(I+be ).

C and, for each z in C, J(o)z is integrable on com-

pact intervals.

is a

function v:

By Theorem 1.4 of [4], if z is in C then there 2 ~.( such that v' + v = J(t)v, v(O) = z. Let t x - 1>O/(I+b) and let u(t) = vet) + ¢o/(l+be ) with v as above. t u' + u = A(u,u) and ufO) = x. Also, lu(t)-1>O/(I+be ) I ,; IX-1>O/(I+b)l o exp(ct) where c =} Ix-¢O/(I+b)1 - (l+3b)/3(l+b)


[0,00)

O.

When b = 0 we have containment in .(2 of the solution for 4 IX-1>ol < 3"' This improves the

o with initial value x where

previous estimate Remark.

Then

IX-1>o I ,; .7085 which was obtained in [3].

In a similar manner,

it can be shown that if b and a are

related by b ~ 0 and 21~1 < 3(I+b) and if Y = a./(l+b)

then we have

this stability result:

I f Ix-yl

is only one function u:

[0,00) ->- .(2 such that u is a solution of (TW)

and ueO) = x.

Moreover,

< 4(3(1+b)-21&'1]/3(1+b)

then there

lu(t)-~/(l+bet) I 4- 0 as t increases.

49

BOL TZMANN EQlJA TIONS IN HILBER T SPACE

§4.

LINEAR OPERATORS WHICH COMMUTE WITH A In this section we demonstrate that there is a two-parameter semigroup of bounded linear operators T b such that a, ACTa , bCx),T a, bCy)) = Ta, bCA(x,y)) for all x and y in a dense set. Al so T b T d = T b bd' a,

C,

a+ c,

We use the following notation: Aa is the linear operator A(;;,.) and N(x) = min{n: x(n) fO} for x f 0 and x in ,[Z. T__h_e_o_r_e_m__4~._1.

Suppose -1 < a < 1.

Then A

is a one-to-one, Uilbert2 a Schmidt operator with IIA a II s 1/(l-a). Furthermore, the non-zero spectrum of A is {lin: n = 1,Z,···} and each eigenvalue has multiplicity 1.

-

Z

Proof. From Lemma 2.2, we have that A is a bounded, linear operator 2 2 a and IIA 11 25 1;;1 = l/(1-a). To see that A (y) f 0 i f y f 0, let a a 1 n = N(y) and note that 0 , ~l x = n+ n+p 1 ,n+p n+p-k 1 ,p-l p-k n+p+l (x n +p + Lk=Oa x n +k )· Thus, x n +p n+p+l Lk=Oa xk n+l ,p-l p-k P L.k=Oa xn+k' And, we see that upon choosing xn ' x is completely oo P¢ determined and is x n Lp= o(p+n)a n n+p Corollary 4.2. If -1 < a < 1 and x f 0 then A(a,x) = AX if and only if there is a nonnegative integer n such that A = l/(n+l) and x oo P¢ cL p= o(p+n)a n n+p for some c f O. Remark.

In a similar manner it can be shown that if x is in ,[2 and

Xo f 0 then Ax ~ A(x,') is a one-to-one, Hilbert-Schmidt operator with I IAx\ I S Ixl. As before, the non-zero spectrum of Ax is {x0/n: n = 1,2,"'} and such members of the spectrum are eigenvalues of multiplicity 1. If Xo = 0, then Ax is quasi-nilpotent; that is, its spectral radius is zero. Theorem 4.3. Let T be a linear operator on D, the span of {¢ p }oop= 0 ' such that A(Tx,Ty) = TA(x,y), for each x and y in D. Then, there

50

M.F. BARNSLEY et al.

are numbers a and S such that

lal

< 1 and T(x) (n)

,n (n)an-PSpx n = 0,1,2, ••• Lp=O P p , Proof.

1fT has the commuting property and is 1 inear, then using the

nonlinear eigenvalue l)rojierty, A(H

H ) = __ 1_ T(w ). In parm' n m+n+l m+n ticular, A(TwO,TwOJ = T(w ) so that either T(¢O) = 0 or, by Theorem O 2.4, T(w ) = & for some a in (-1,1). Furthermore, A(T¢n,T¢O) O ~1 T(¢ ) so that j f T(w O) = 0 then T(¢ ) = 0 for all nand T" D. n+ n . n 1 If T t 0 and T(w ) = &, then A(H ,&) = --1 T(w ) so that, by Coroln n+ n O

snLp=O ,00 (p+n)aPw n p+n

lary 4 2 T(w) = ., n

°

= B ,'" (P)aP-nw for some senLp=n n p

quence {s }"" and lal < 1. To determine {S }oo_O' we examine n n= n n,HI) (n+l) = ~7 T(q, l)(n+I) = A(H ,H ) (n+l) = n n+~ n+ n l _1_ ,n+l Tew ) '1'(q,) But n+2 Lk=O n n+l-k 1 k .

A(H

0 if k = () k-n

T(W ) (k) 1 and

B1(~)a

Ia I

Ibl

a)

If

If lal + Ibl

+

< 1 then T

k

n +1

,;

(n+l-k)a l-k if k = 0 or 1. n

Suppose a and b are numbers and

b)

,;

0 if k > 1

T(Wn)n+l-k

Bn

Theorem 4.4.

if 1

a,b

lal

< 1.

is Hilbert-Schmidt.

~ 1 then Ta,b 1S bounded and

IITa , b 1l2 ,; l/[l-laIClal+lbl)] c)

If lal

+ Ibl

> 1 then there is x in (2 such that x is not in the

domain of T a, b Proof.

(a)

To see that T

a,

,00 ,n (( n ) an-PbP) 2 Ln=OLp=D p

b is Hilbert-Schmidt, sum: ,;

C,n Cn) lal n - P lbI P )2 L.n=O Lp=O p

\,00

00 2n Ln=o(lal+lbl)
- 0 all,1•

of c' + c = In oa c c c (0) = a i.n .£.2. Or, as in Theoreml.l, n n p= np n-p p' n ' n dU y 2 8t (t,x) + uet,x) = ! dy! dz k(z,y)u(t,y-z)u(t,z) with u(O,x) in L , x 0 for appropriate choices of k.

52

M.P. BARNSLEY et al.

REFERENCES 1.

M. F. Barnsley and H. Cornille, General Solution of a Boltzmann Equation and the Formation of Maxwellian Tails, Proc. Royal Soc. London A, 374 (1981), 371-400.

2.

M. F. Barnsley and H. Cornille, On a Class of Solutions of the Krook-Tjon-Wu Model of the Boltzmann Equation, J. Math. Phys. 21 (1980), 1176-1193.

3.

M. F. Barns1ey, J. V. Herod, V. V. Jory, and G. B. Passty, The Tjon-Wu Equation in Banach Space Settings, Journal of Functional Analysis (To appear).

4.

H. Brezis, OperateursMaximaux Monotones, North Holland Publishing Company, Amsterdam, 1973.

S.

J. A. Tjon and T. T. Wu, Numerical Aspects of the Approach to a Maxwellian Distribution, Phys. Rev. A 19 (1979), 883-888.

Spectral Theory of Differential Operators I.w. Knowles and R. T. Lewis (eds.) © North-Holland Publishing Company, 1981

SOME PARTIAL DIFFERENTIAL OPERATORS WITH DISCRETE SPECTRA

v.

John

Baxley

Department of Mathematics Wake Forest University Winston-Salem, North Carolina U.S.A.

We study selfadjoint realizations of the formal differential -1

[(Plu) + (P2u ) 1 in the weighted Hilbert x x Y y space Lm (S'l) where rl is the square domain (0,1) x (0,1) . Assuming

operator Tu 2

=

-m

m, PI' P2 are positive and reasonably smooth and that singularities of T occur only along the boundaries x

=

0 or y

=

0, a variety of

strictly positiveselfadjoint realizations of T are constructed, each of which, with a further integrability condition on the coefficients, has a discrete spectrum.

1.

Let T be the formal differential operator

(1)

1

TU

If rl is a domain

m

+ (P2 uy ) Y 1.

[(Plu)

x x

in~2, this formal operator may give rise to a variety of self-

adjoint operators in the weighted Hilbert space L2(rl) consisting of all m

measurable complex-valued functions u defined on rl for which Ilul ~

=

(II

I ul

2

1/2

)

m dxdy

rl


O and PDf 0, and that

T is

formally symmetric of

lower order. Then the equation Sy=Tz may be equivalently written (see [31 section 2) u' + [- c *

(1 • 3 )

A

HJ u C

=

[0 0Jv'

+

G* 0

HG OJ [ B+CG 0 v

where A,B,C,G,H are mXm matrix-valued and A,;:O, 1m G C 1m H. Put s a [-:* where

q

~)

, t

a

[

G~H

HG B+CG+G*C*

J

, k

H~O,

B*=B and

[~ ~J

, q =

[~ ~J

is chosen so that qH is the orthogonal projection on 1m H.

Then (1.3) is the corresponding Q-hermitean relation, u and v are in

cl

and the system is left-definite according to our definition (with j 2 Jl:P.I/ )1 ), but ("')S,J is not positive ]

(u,u)S J coinciding with on all' of C1 (1) • B~~~r~~

The spectral theory in [7J is carried out for systems of the form (1.3) with G=O and B satisfying -pA,:SB,:SpA for some p ElL

64

CHRISTER BENNEWITZ

2,SPECTRAL THEORY Let ( " ' ) J denote ("')T,J in the right-definite and (·,·)S J in the 1 be C (I) in the right-definit~ and as left-definite case and let

cl

in section 1 in the left-definite case. Considering the part of C*1 giving a finite value to (u,u)I and introducing the quotient with respect to elements with vanishing norms we obtain after completion a Hilbert space H with norm IUl 1

r

1

Eloc = {(u,U)E C*xC* EI

1:1

{(u,u) E E

D\

1:1

{(u,\u) EEl}

E\

D\

+ D\

=

~r

Su=:Tu}

loc

for 1m \

t 0

BJ(U,V) =-i(u,v)J-(u,v)J) EO

I

Br(U,E ) = O} I One might view EI as the maximal and EO the minimal relation associated with S and T in the norm I . I I The basis for the spectral 1:1

{UEE

for U=(u,u), V=(v,v)

I

theory is then given by

!l:!§:9!:§JE':' 1 • Er = EO .j. EA

as a direct sum

1

2. For VE C* with Ivlr. y' subject to (1. 2), is not densely defined so that L * does not

the conditions

exist as an operator. 0(L *L).

Thus,no meaning is assigned to L *L or to

The same difficulty would hold also for

Theorem 4.

(Fan, Taussky, and Todd [19J) If y

E:

2 W ,2(O,n),

n

y' (0) = y' (n) and f y = 0, then -0

with equality if and only if y

Theorem 5. y

E:

cos t.

(cf. Everitt [17]) Let p be

w1 ,ZCO,2n) 2n f

o

~

given positive integer,

and satisfy the conditions 2n ycosnt=f

yeO)

ysinnt=O,n

O, ... , p - l ,

0 y(2n).

Then

with equality if and only if y A different problem in the application of Proposition I is apparent for the following inequality relating the minimum of a Dirichlet functional to the infimum of the spectrum of an associated s.a. differential operator. Theorem 6. Suppose

-00

< a < b
0, p

-1

. and q locally lntegra-

ble functions on [a,b) (i.e., in LfocCa,b)). Further suppose q is essentially bounded below. Define T: L 2 Ca,b) + L 2 (a,b) by M[f] = C-l)n(pf(n)) (n) + qf on the domain of the maximal operator

72

RICHARD C. BROWN

T+(M) determined by f such that (pf(n))(i)(a) = 0, i = 0, ... , n - 1. Assume that the minimal operator TO(M) is limit-no Then (1. 3)

b

()

b

2

f ply n 12 + f q Iy I

a

a

b 2 > 110 fly I , a

inf

)1

(J

(Tl ,

0

for all y in D: = {y E L2 (a,b): y(n-l) E AC and the integrals on the reft of (l~) are absolutely convergent}. 1 EquaTity holds Ifty-rs-an eigenfunction corresponding to 110' If 110 ¢ a p ' equari~holds if and only if y = O. But there is a sequence such that IIYkl1 = 1 and 0. f bpl'y(kn ) I 2 + fb qly k 12 - 11 0 fb ly k 12 ~ ~ a a a Certainly Theorem 6 reminds us of Proposition 1. is "what is L?"

The question here

One purpose of this paper will be to give an extension of Proposition 1 which is adequate for Theorems 3-6 and other inequali ties as well. We proceed to outline the contents of the paper. The desired extension of Proposition 1 - "Proposition la" - is presented in Section 2 using a theory of linear relations in Hilbert space developed in recent years by Cgddington [15),[16) and also earlier by von Neumann [27), Krasno~erskfi[25), and Arens [51. The proofs of Theorems 3-6 and of certain additional corollaries will be given in Section 3. Section 4 discusses some extensions of the theory to inequalities with interior point boundary conditions, andsketches the relation of some of our results to those of others. The paper is intended to be self-contained "almost everywhere" in that the significant arguments are sketched in some detail or in the occasional instance where this is not possible full references are given. (The only exception will be Theorem 8, Section 4.) 2. THE SPECTRAL THEORY OF LINEAR RELATIONS Let H, H' be complex Hilbert spaces. A linear relation L is a setvalued mapping on DeL) c H to H' whose graph GeL) is a subspace of H x H'. (We find i t useful to distinguish between Land G(L) although this need not be done. One can identify the relation with its graph and speak directly of subspaces as is done for example in [15) or [16).) L is closed if and only i f G(L) is closed in the usual norm topolo£y of H x H'. L is normally solvable if it is both closed and has closed range. For a E D(L) the image set in R(L) will be denoted by L(a); an arbitrary member of this set will be signified by La. We define IIL(a) II by dist(La + L(O)): = inf{IILa + yll: y E L(O)}, i.e., as the norm of

WIRTINGER AND DIRICHLET INEQUALITIES AND SPECTRAL THEOR Y

73

an element in H'/L(O). Supposing L is closed, L(O) is a closed subspace of R(L) and S E L(a) if and only if S ~ a mod L(O). The nullspace N(t) of L: = L-lL(O) ~ {a E D(L): (a,O) E G(L)}. Given relations L,M we define LoM such that G(LoM): = {(a,S): (a,y) E G(M); (y,S) E G(L)}. The adjoint L* of L has graph ((a,S): [y,a] - [x,S] =O,II(x,y) E G(L)}. Clearly L*(O) = D(L)·J.··. Let L be defined in H. Then A E pel) if (L - AI)-l is a bounded operator from H/L(O) to H. o(L) is the complement of pel). A E opeL) and ~ is an eigenfunction corresponding to A if (~, A~) E G(L), equivalently if A~ = L~ mod L(O). There are close parallels ~etween the adjoint and spectral theory of operators and that of relations. For instance the Fredholm alternatives and closed range theorem are true in both cases (cf. [5 ],[16) , [27) for detai Is) . We now give a generalization of Proposition 1 adequate for inequalities on nondense domains. Proposi tiOH la. Let H, H' be Hilbert spaces and L: H ..,. H' a nondensely deflned normally solvable operator. Then L*L is a s.a. ii:ClTIiiaTIy solvable relation and IIY II 2 ].1(jl/2IILy II where ]..10: = inf o(L*L) . .!.i]..lo E 0p(L*L) equality is attained at 1/1 if and only if 1/1 is an eigenfunction of L*L. .!.i].10 t 0p(L*L) then equality holds if and only if y = 0, --but ----there exists -a sequence .¢n E D(L*L) ----- -- --- ---- - wi th II ¢n II

= 1

such that lim II Un 112 - ].1~l

=

0 as n ..,.

00.

L- l is defined and bounded by the closed graph theorem. Let ~orthogonal projection onto R(L). Then IIL-lpil = Ilvlll and L-lp maps Il' onto D(L). Consider(L-lp)*. Since O:-lp)* = (L-lp2)* = P(L-lp)*, R(L-lp)* c ReL). Let y E D(L) and -1 * -1 * * z E H. Then,[(L P) z, Ly] - [z,y] = 0 so that ((L P) z,z) E G(L ); thus (L -1 1') * maps H into D(L * ). Set T: = L-1 pel -1 P) * . T is s.a. Set S: {(Tz, z + y): z E H' , Y E L* (D)}. Routine computations show that S L*L and that S is s.a. Thus L*L is s.a. We next show o(L*L) = o(T)-l. This means ]..10 is real and positive since T is s.a. Let Q be orthogonal projection on D(L). We claim that T (Q L*L)-l. To see this let y E DeL) and z E H. Then ([T Q L*Ly,z]) = [(L-lp)*Q L*Ly, (L-lp)*z] = [L*Ly, Q L-l(L-lp)*z] = [Ly, (L-lp)*z] Proof.

[y,z], so that y = T Q L*L),. I t follows that oCT) = o(Q L*L)-l. Let ]..I be a complex number and Z E D(L). It is easily checked that II(L * L z - ]..I z) /L * (0) II = II Q L*Lz - ]..IZ II. This fact implies from our definitions that p(L*L) = p(Q L*L) and that o(L*L) = o(Q L*L). Consequently oCT) = o(L*L)-l. To complete the proof, we observe that

=

74

110

RICHARD C. BROWN

- Yz

i1.

= IlL·· 1 II = IIL- 1 pil =

k

I!TI12~

0/T)2. But Further since o(L*L)

ll~l E oCT) llO E o(L*L).

>

0,

llC/ sup 0(T) 110 = inf o(L"'L). Finally, the statements concerniEg equality follow from standard theory (cf. [24], p.234). Corollary 1. Let the hypotheses of Proposition la be satisfied. Suppose also L has ~ compact partial inverse. Then,110 is the least positive eigenvalue of L*L. ~ality is attained by ~ E DeL) if and only i f ~ is an eigenfunction of L *1. Corollary 2.

Suppose L satisfies the hypotheses of Corollary 1. L on Dn: = {y E DeL): [y,E;i) = 0, i =1, ... , n - l } where {(.} are the first n - 1 eigenfunctions of L*L. Then

IJeTII1e---r:;c ~

IlL n II

-=" n-r-where" is n1

the n

th

-*

.

eIgenvalue of L L.

Proof. It can be shown Ccf. [11)) that G(LI~) = {(y, L*y + 1jJ): [y, t:il = 0, 1jJ = L c j i;i} where the c i are arbitrary compl~x parameters. Hence the eigenvalue problem is L*Ly = "y + l)!; [y, E;i 1 = 0, i = l, ... ,n - 1. This, however, implies l)! = 0, so that standard theory applies to show that 110 = "n· 3.

APPLICATIONS

We now show how PropOSition 1 or la applies to the theorems of Section 1. With the exception of Theorem 6 the fact that a given L is normally so~vable and has a compact inverse as well as the structure of L can be read off from theory in [10]or[14). Proof of Theorem 3. Define L by y' on 1 2 211 D: = {y E W ' (0,211): yeO) = y(211); (, y dt = O. Then, l G(L*) = {(y, -y' + ¢): y E W ,2(0,211): yeO) = y(211); ¢ an arbitrary complex parameter}. By Corollary 1 the best constant in Wirtinger's inequality is 1 for -y " y (0) (3.1)

y' (0) f211 y

a

"y

+

¢

y (2iT) , Y (211) 0

for some complex ¢. Integration and use of the boundary conditions in (3.1) shows that ¢ = o. Therefore, 1 = 1 with an eigenmanifold spanned by sin t and cos t.

75

WIRTINGHR AND DIRICHLET INEQUALITIES AND SPECTRAL THHOR Y

2 Proof of Theorem 4. Define L by v" on the subspace of W ,2(0,'Tf) satIsfYIng the boundary condition (3.1). L* is given by y 1--+ y" + ¢ with y' (0) = y' (11). y (i v) y'( 0) Y (iii) (0) 11 J y

The eigenvalue problem is icy + ¢ y ' C1I ) Y (ii i) (11)

O.

° The rest of the proof parallels that of Theorem 3. Proof of Theorem 5.

This is an immediate application of Corollary

~

Proof of Theorem 6.

Here Proposition 1 is sufficient but L needs to be carefully defined. Define L: L 2 (a,b) ~ LZ(a,b) x LZ(a,b) by Pl/zy(n) ) , y (

y I---'"

( (q+d)

D

liZ y

where d is such that q + d > ( > 0. Clearly L is densely defined and 1-1. It is straightfor~ard to show that L is closed. Further . 1/2 (n) 1/2 L has closed range. For If p Yk -7 U and (q + d) Yk -7 V our

-liZ

choice of d guarantees that Yk -7 v(q + d) . But since the operator y I---'" pl/Zy(n) on = {y ( L 2 (a,b): y(n-l) (AC; p1/2y(n)}

D:

is closed (this follows by the hypothesis on p), = u. Moreover (q + d)1/2(v(q + d)-l/2)= v

p1/2(v(q + d)-l/2)(n)

so that (u,v) (R(L). Define L+: LZ(a,b) x LZ(a.b) -7 LZ(a,b) by +( ) -, n 1I Z (n) liZ L Z I' Z Z =: l -1) (J1 Z 1) + (q + d) Z on l D*: = {(zl'zZ): ( p /2 Z1 ln-l)( AC;(pl / 2 z1 )(i (a) = 0, i = 0, ... , n - 1, [y, (b') = o}. [[ere [y, ~] (b-) is a form discovered by integrat-

J

zi

ing (-1) n (pl/2 z/n) y by parts. is to show that L+* = 1.

The next step, which is not difficult,

Therefore, L * = ~ L (this is an operator

since L is densely defined.) It turns out further that if OJ * + .. 1 L* L IS . q ( L (a,b), L = L. By ProposItIon, s.a. Furt h er

*

---=F

-+-

L L := L L = L L.

*

+

Now L L c Td:=T + d whence L L c T . d the limit-n condition Tdis s.a. I t follows that L*L

*.

.,

)lod: = inf a(T d )· Because L L IS posItIve )lO,d > 0, Proposition 1 gives

Since we have Td . Let Applying

RICHARD C. BROWN

76

~bpIY'IZ+

(q + d) lyl 2 > )10d llylZ

.

By the spectral mapping theorem )10d - d = )10 = inf 0(T). discussion of equality also follows from Proposition 1. -1

The

1

Corollary 3. Suppose b < 00, p , q E L (a,b). Then Theorem 6 is true provided functions f in DC!) satisfy c;f(n))(i)(a) = (p f(n))CIT(b) = 0, i = 1, ... , n - 1. Moreover, )10 is an eigenvalue and equality in (1.3) is attained CIt an eigenfunction. Proof. We approximate T by sequence of operators Tn such that qn is essentiall'y bounded below and 1!- -y" on the subspace M of W2 ,2(0,211) satisfying the orthogonality condition J211 y = 0; y and y' also satisfy periodic boundary conditions. This gaarantees that T is defined in M. Thus T is a reduced operator in the sense of Akhiezer and Glazman ([1] ,p. 82) and is s. a. I t has first eigenvalue A = 1 with eigenmanifold spanned by sin t, cos t. The numerical range inequality and integration by parts give Wirtinger's inequality on D(T). The inequality is extended to the larger domain D(L) by an approximation technique. This method however seems difficult to generalize to other Wirtinger-like inequalities, e.g., Theorems 3-5 above. By contrast our method gives the same equations as a calculus of variations approach and thus can be viewed either as a spectral interpretation of this approach or as a rigorous justification of it. Further details and other results are given in [12]. In the past decade much work has also been done on Dirichlet functional inequalities in the case n = 1. See, for example, Bradley and Everitt [7],[8], Amos and Everitt [2 -4], Sears and Wray [28], and Everitt and Wray [18J. Additionall~ material relating to the higher order case but in a different setting can be found in 19] and 122]. The methods and hypotheses of these papers, however,

78

RICHARD C. BROWN

differ from our own. Implicit in much of this work is the discovery that the domain on which the inequality is valid is the domain of the square root of Td . This fact also follows from our approach; indeed D(L) = IX/I7'L) for all the inequalities considered in this paper as is clear from Kato [23], Ch. 6.7 (2.22) p.334. We have also not considered the weight function case here (as is done in [18]). This case produces an inequality of the form

-1

b 2 f plf' 12 + qlfl > a , ,

b 2 f wlfl °a

jJ

where M[f]: = w [- Cpy) + qy] with q > -kw. But such an extension by our approach would be simple. Also~ our method works well for more complicated boundary conditions than considered explicitly here. On the other hand it does not yield inequalities like (1.1) or (1.2) of [18]. For further details and some extensions see [13]. 1.

AC means local absolute continuity in the singular case. REFERENCES

[1] Akhiezer, N.I. and Glazman, I.M., Theory of Linear Operators in Hilbert Space, Vol. I (Ungar, New York, 1961). [2] Amos, R.J. and Everitt, W.N., On a quadratic integral inequality, Proc. Roy. Soc. Edinburgh,Sect. A 78 (1978) 241-256. [3] Amos, R.J. and Everitt, W.N., On integral inequalities associated with ordinary regular differential expressions: Eckhaus, W. and Jager, de F.M., (eds.), Differential Equations and Applications (North-Holland, Amsterdam, 1978). [4] Amos, R.J. and Everitt, W.N., On integral inequalities and compact embeddings associated with ordinary differential expressions, Arch. Rational Mech.Anal. 71 (1979) 15-40. [5] Arens, R., Operational calculus of linear relations, Pacific J. Math. 11 (1961) 9-23. [6] Beckenbach, E.F. and Bellman, R., Inequalities (SpringerVerlag, 1961). [7] Bradley, J.S. and Everitt, W.N., Inequalities associated with regular and singular problems in the calculus of variations, Trans. Alner. Math. Soc. 182 (1973) 303-321. [8] Bradley, J.S. and Everitt, W.N., A singular integral inequality on a bounded interval, Proc. Amer. Math. Soc. 61 (1976) 29-35. [9] Bradley, J.S., Hinton, D.B., and Kauffman, R.M., On the minimization of singular quadratic functionals, preprint. ~O] Brown, R.C., Duality theory for nth order differential operators under Stieltjes boundary conditions II: Nonsmooth coefficients and nonsingular measures, Ann. di Mat. pura ed appl., 105 (1975) 14l-l70. [11] Brown, R.C., Notes on generalized boundary value problems in Banach spaces, I adjoint and extension theory, Pacific J. Math., 85 (1979) 295-322.

lVIRTINGER AND DIRICHLET INEQUALITIES AND SPECTRAL THEOR Y

79

[12] Brown, R.C., Wirtinger's inequality and the spectral theory of linear relations, preprint. [13] Brown, R.C., The minimization of a Dirichlet functional as a problem of operator theory, preprint. [14] Brown, R.C. and Krall, A.M., Adjoints of Stieltjes boundary value problems, Czech. Math. J., 27 (1977) 119-131. [15] Coddington, E. C., Spectral theory of ordinary differential operators, in: Dold, A. and Lckmann, B. (eds.), Spectral Theory and Differential Equations (Lecture Notes in Mathematics #44S, Springer-Verlag, Berlin, 1975). [16] Coddington, E.C., Adjoint subspaces in Banach spaces with applications to ordinary differential subspaces, Ann. di Mat. pura ed appl., llS (197S) I-lIS. [17] Everitt, W.N., Spectral theory of the Wirtinger inequality, in: Dold, A. and Eckmann, B. (eds.), Ordinary and Partial Differential Equations, Dundee 1976 (Lecture Notes in Mathematics #564, Springer-Verlag, Berlin, 1976). [18] Everitt, W.N. and Wray, S.D., A singular spectral identity and equality involving the Dirichlet functional, preprint. [19] Fan, K., Taussky, 0., and Todd, J., Discrete analogs of inequalities of Wirtinger, Monatschefte fUr Mathematik 59 (195~ 73-90. [20] Gohberg, I.C. and Krein, M.G., Introduction to the Theory of Linear Non-selfadjoint Operators (Translations of Mathematical Monographs Vol. IS, American Mathematical Society, Rhode ISland, 1969). [21] Hardy, G.H., Littlewood, J.E., and Palya, G., Inequalities (Cambridge University Press, 1967). [22] Hinton, D.B., Eigenfunction expansions and spectral matrices of singular differential operators, Proc. Roy. Soc. Edinbufgh, Sect. A. 80 (1978) 289-30S. [23] Kato, T., Perturbation Theory for Linear Operators (SpringerVerlag, Berlin, 1966). [24] Krall, A.M., Linear Methods of Applied Analysis, (Addison-Wesley, Reading, Mass. 1973). [25] Krasnose~skiI , M.A., On the extension of Hermetian operators with a nondense domain of definition, Doklady Akad. Nauk SSR (N.S.) 59 (1948) 13-16 (Russian). [26] Mitrinovic', D.S., Analytic Inequalities (Springer-Verlag, Berlin, 1970). [27] Neumann, J. von, Functional Operators, 1., Annals of Math. Studies, No. 21 (Princeton University Press, Princeton, 1950). [28] Sears, D.B. and Wray, S.D., An inequality of C.R. Putnam involving a Dirichlet functional, Proc. Roy Soc. Edinburgh, Sect. A. IS (1975/76) 199-207.

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Spectral Theory of Differential Operators I. W. Knowles and R. T. Lewis (eds.) © North-Holland Publishing Company. 1981

A SURVEY OF SOME RECENT RESULTS IN TRANSMUTATION

Robert Carroll University of Illinois at Champaign-Urbana

I. Introduction. This is a very abbreviated survey of some work done in the past few years on the theme of transmutation (Sections 5 and 6 represent new material).

We shall omit most references, for brevity (they can usually be found

in the bibliographies to our papers) and all of this material will appear organized together in a new book [ 101 which we are preparing now.

We consider only

second order differential operators of the form Qu = (~Qu')' /~Q here (plus suitable perturbations) under two kinds of hypotheses:

(A) Q is modeled on the radial

Laplace-Beltrami operator in a noncompact rank one symmetric space (~Q

+ x or sh + x ch + x for example) or (B) ~Q E c , 0

2m l

2a l

[0,00),

~Q ~ ~Q(oo)

sh

, set Q(D) = Q(D)

+

{~Q("'/~Q)'}' + P~'"

28 l

l

2m+l

h P 2 were P Q

Q

1'2 l'1m uA' /Au Q

Q

as x ~

00

- ~ Q < A < 00 on We

'*'" with formal adjoint Q

(the nonselfadjoint formulation is deliberate and useful).

Here P and

0 are

P = 0)

Q (B: P + 0)

We

if OB = BP acting on suit-

of the form above and B will usually be an inte-

gral operator with distribution kernel. ized translation when

or

< a


Y~ ~

2

(x _/)n-m-3/2 dy where

Y: X(Dx/X)n~ g(y)y2m+l.

~

for -!z < m < n-lz and suitable g, (yQ(x,y)g(y)

n l 11T/2 - r(m+1)f'(n-m-lz).

We will say more about SQ

and YQ later. Let us briefly indicate some techniques of studying Parseval formulas

r 22;29]

in our framework which are based on b Let fey)

(B *O(y)

~

~

(cf. also [12;13;26]). -

V and g(x)

(y(x,y),f(x)

(B *g)(x)

~

~

(S(y,x),g(y).

Qf(A) ~ Pf(A) and Pg(A) ~ Qg(\) where Pf(A) ~ ( f(x) ,~~ (x) -k

~

Note also that B ~

QP and B*

~

PQ where QG(y)

Lemma~.

Then

and Qg(\) ~ ( g(y), (G(A),Q\Q(y)w and PF(x)

~

P

(F(A),Q\ (x)v. - Now one wants to determine a Parseval formula of the form

-~ -!2 (II) (R,PfPg\ ~ (6 p f,~p g) for Pu ~ (~pu')'/~p - q(x)u where q is a suitable po-

tential and R is a generalized spectral function. m

First let B: P

m

change in notation - here v y

form (t) Txop(x)

~

(R

"u

~

D2 ~ Q

+

Pm and w

P

w

~ P

P

(A)'~A (x)~\

0,

Q).

Q (cf. Example 2.2 but note the One tries to find rYe (x) in the x P

(Y)w ,.here dw

(~;lzf,~;\) ~ (g(y),(T~Op(x),f(x)) ~ Y

ul

~

(2/TI)d\.

KP to get (Bop)(y) ~ KPPR

w

P

w

Now B

Operate on this with B

~

Thus RW(\)

BQ from Example 2.2 so Y Q

~

Setting

~ kR w (since K- l ~ PPQ and Q-l ~ k); this holds when-

~~ (no spectral considerations).

Cos Ay dy.

Then for suitable f,g

(RU10),PfPg)w which reduces to (II).

0 now in (t) we have apex) ~ (R (A)'~A (x)w ~ f-lR.

ever B~~

be the

m

We connect P

model of Example 2.2. transmutations.

To illustrate let P

~

ker B.

Some nontrivial but rou-

tine calculations with distributions show (as is apparent on other grounds) that

Now having "discovered" Ro we connect P Again try (1") where TY x

"u

are as in Example 2.2).

2

(m -~)/x

2

and x

m+lz

"u

1
-)}.p~(y)dA. -

=

o

Similarly an integral equation F(>-)

I;

=

f(y)i(y)dy (assume CQ(-A) is

not known) can be reduced to a Volterra equation if we can compute AQ(y,x) (1/2n)[:

~(y)exp(-iAX)d>- where by known triangularity AQ(y,x)

Setting T(x)

'"

0

very rapidly decreasing q, SeA) ImA > 0 and fey)

=

(1/2TI)!:

=

x.

>

2 _A u and for suitable

=

CQ(A)/CQ(-A) will be analytic in a halfplane

F(A)~A (y)dA.

5. Elliptic transmutation.

The transmutations P

~

by spectral pairings required basically that the spectra of Consider now - Example 5.1. Let P = D2 =

P and

Q = _D

2

2 For Q consider _D W = _AZW with W(O)

with A E [0,00). AX

0 for y

= (1/2n)I'" F(A)exp(-iAx)dA one obtains IX AQ(y,x)f(y)dy = T(x).

Still another approach refers to equations u" - q(x)u

e-

=

Set Qf(A) = f(A) =

Z ZY/TI{x +y2}.

Also

B 1, or e-tH_e- t Ho is trace class, where Ho and H are respectively -A and -A with boundary conditions. Our results in completeness are based in the verification that f(H)(H-H ) f(H ) is trace class, for a suitable chosen function f. This operator is easi9y ex_ o pressed via the Green's formula, in terms of the obstacle and boundary conditions. As an advantage we do not need to obtain estimates in the resolvent or heat kernels. We obtain the required bounds just by Sobolev imbedding theorem [26]. Moreover our proof does not appeal to the invariance prinCiple of wave operators. In the following section we state our results in existellce and completeness of wave operators. A detailed version will appear in [27] . The asymptotic expansion of phase shifts will be considered elsewhere. II The Results Let Qe' the exterior domain, be an open set. The obstable, Q., is the complement of Qe,Q; = IR n-r2 e • We will not assume that "i is bounded. We ~ill consider the situation where ". has a bounded part contained in a ball of radius R, where no regularity assumptlon will be imposed, and an unbounded part, contained in the complement of the ball, satisfying mild regularity assumptions.

95

SPbCTRAL THEOR Y AND UNBOUNDED ORSTACLb SCA TTERING

We assume that (2.1 ) aile = all e ,l u dll e ,2 ' where dlle 1 is contained in a ball of radius R, and dQ e, 2 is contained in the compl ement of the ba 11. Denote (2.2) Il ={XUl I Ixl>R}. R We will impose our regularity assumptions in dll R, which amounts to require regularity of all 2' We assume that

N e,

where

IlR =k~l Il R ,k Il ,k are open sets with the R Il ,j n Il ,k = cp , j "f k ,

(2.3) property (2.4)

R

R

and where each ~R k has a bounded trace operator on dllR k' We will consider situations where' there is no global trace operator, but there is a trace operator for each of the connected components of Il R. For example an obstable with the set of directions along which it is unbounded of measure zero, or an unbounded surface. Let us take polar coordinates (p,w), PElR+, wcs n- 1 , onlR n . Denote E= {

W

s Sn-1

1

j {

wn }~=1 ' {'\n}~=l ' wn E Sn-1,

wn -> W, An s lR + , An -> and Anwn £: dlle} , (2.5) that is to say E is the closure of the set of directions along which the obstacle is unbounded. Our ~tin condition for existence of wav 0Rerator~ is that E is of measure zero in Sn . Denote by J the operator from L2(lR ) to L (Il e ) given by multiplication by the characteristic function of 11 . And let Ho be the selfadjoint realization of -/'; in LL(JRn). e 00

,

Theorem I Let H be a selfadjoint operator ~n L2(11 ) such that for every ¢sD(H) we have £: H (Il ), JRH¢ = -/';cp in L (Il R). eand 2 R R

¢11l

IIcpIlH (Il ) ,;;; K(IIH cpll + 1I¢1I), (2.6) 2 R for some constant K. Suppos~ that E has measure zero and that HI (OR k) has a

bounded trace operator on L (d~R,k)' for 1 ,;;; k ,;;; N. Finally we assume that for some M > 0

J 31l

(l+lxl )-M dS
0,


0, with values in Cn. Then there exists a unique periodic solution of the system (1), of period T. The Fourier series of the solution is absolutely convergent. Indeed, for any periodic function, regardless of the period, the condition (9) is satisfied. The Fourier exponents of a periodic function form an arithmetic progression. Theorem 2. fying (3).

Consider the system (1'), with t j , Aj , j = 1,2, ... , and B satisAssume that f E B2 (R,C n). If the condition (10) is satisfied, and 1 2 ( 1+t ) II B( t) 112~ L (R+) ,

( 12)

then system (1') has a unique B2-almost periodic solution. Remark.

The Fourier-Stieltjes transform A(is) is now

( 13)

A(is) =

I A. exp(-it.s)

j=O

J

J

+

JooB(t)eXP(-itS)dt, 0

s

E

R.

From Theorems 1 and 2, one can see that an appropriate kind of almost periodicity for the function f implies the same, or another kind of almost periodicity for the solution. Nevertheless, the following problem is still open: does f ~ AP(R,C n) imply the existence of a solution x ~ AP(R,C n)? Another type of problem, concerning almost periodicity of solutions (1), would be the following Bohr-Neugebauer type of problem: prove bounded solution of (1), with f E AP(R,C n ), is also in AP(R,C n ). without using hypothesis (10). See [9] for differential equations in

of the system that any Of course, Banach spaces.

PROOF OF MAIN RESULTS Before we can prove Theorems 1 and 2, we will establish the following Lemma. Consider the system (1), with f(t) = b exp(iAt), b ~ cn, A ~ R. If condition (10) is satisfied, then there exists a unique solution x(t) = hexp(iH), n hE C •

Proof. One proceeds by direct substitution of f and x in the system (1). linear algebraic system from which h has to be determined is (14 )

[iAI - A(iA)]h

=

The

b,

and it has unique solution by virtue of condition (10). Corollary. ( 15)

There exists a constant

K > 0,

depending only upon A(s).

Ih I ~ KI b I·

Moreover, the following estimate holds true:

such that

ALMaS]' PERIODIC SOLU110NS l'OR INHNITE DELA Y SYSTEMS

103

(16)

where the matrix norm is the Euclidean one. The proof follows at once if one takes into account that the entries of the inverse matrix [iAl - A(iA)]-l are rational functions of A, with coefficients that are bounded on the real axis, and the degrees of numerator and denominator are respectively (n-l) and n. Moreover the polynomial in the denominator does not vanish on the real axis, and its leading term is (i:\)n. Proof of Theorem 1. From the Lemma and Corollary, one easily finds out that for each (vector valued) trigonometric polynomial m

f(t) = L bk exp(iAkt),

( 17)

1

there exists a unique solution of (1) that can be represented as m

(18)

x(t)

L hk

exp(iAkt),

1

where coefficients satisfy the following inequality (see (15)): (18) By

1·1 one denotes the Eucl i dean norm for vectors in Cn .

Let us now assume that the function

f

in (1) is an element of APabs(R,C n ):

(20)

If one denotes by fm(t) the trigonometric polynomial which is the sum of the first m terms· in the series of f(t), then there exists a unique solution xm(t) of the system (1), with fm(t) as f(t), representable in the form (18): (21)

with

h , k

k

1,2, ... ,

given by

(22)

As seen above, inequality (19) holds true for any m, m = 1,2, ... , and this n implies the convergence in APabs(R,C ) of the sequence {xm}. Therefore, there exists an element x(t) in APabs(R,C n ) which satisfies the system (1), for f(t) given by (17).

104

Assume now that

CONSTANTIN CORDUNEANU

f

E

AP(R,C n ),

with Fourier exponents such that (9) holds true:

It is obvious that an almost periodic solution of (1), if any such solution exists, should have as Fourier series

(24) where the coefficients hk are given by formulas (22). We assume here that x E AP(R,C n ), because the more general assumption x E B2 (R,C n) does not guarantee the convergence of the integral occurring in the right hand side of (1). But taking (16) into account, we conclude that M > 0 exists, such that (25)

This inequality shows that (26)

I Ihkl

k=l

0 and values in Cn , into itself, and satisfies the following Lipschitz condition: (31)

I fx - fy I ~ LI x - y I '

106

CONSTANTIN CORDUNEANU

where the norm is the supremum norm in the space of periodic functions. If L is small enough, then the system (30) has a unique periodic solution of period T. The proof can be easily carried out, relying on the Corollary of Theorem 1, and the Banach fixed pOint theorem. First, an inequality of the form Ixl < Klfl has to be established, where x represents the unique periodic solution, of period T, to the linear system (1). Then one can use the closed graph theorem to prove the continuity of the operator f + x. Remark. The problem of existence of periodic solutions to the system (1), under different assumptions, has been discussed in detail in [6J. REFERENCES [1 J Alexiades, V., Almost periodic solutions of an integrodifferential system

with infinite delay (to appear). Besicovitch, A. S., Almost Periodic Functions (Dover Publications, Inc., New York, 1954). [3J Corduneanu, C., Almost Periodic Functions (John Wiley & Sons, Inc., New York, 1968). [4J Corduneanu, C., Recent contributions to the theory of differential systems with infinite delay (Institut de Mathematiques, Universite de Louvain, Vander, Louvain, 1976). [5J Corduneanu, C. and Lakshmikantham, V., Equations with infinite delay: a survey, J. Nonlinear Analysis 4 (1980) 831-877. [6J Cushing, J. M., Integro-Differential Equations and Delay Models in Population Dynamics (Springer, Berlin, 1977). [7J Daleckii, Yu. C. and Krein, M. G., Stability of Solutions of Differential Equations in Banach Spaces (AMS Translations, Providence, 1974). [8] Hino, Y., Almost periodic solutions of functional differential equations with infinite retardation, Tohoku Math. J. 32 (1980) 525-530. [9J Zaidman, S., Solutions presque periodiques des equations differentielles abstraites, Enseignement Mathematique 24 (1978) 87-110. [lOJ Zamfirescu, T.• Most monotone functions are singular, Amer. Math. Monthly (1981) 47-49. [2J

* Research partially supported by U. S. Army Research Grant No. DAAG29-80-C-0060.

Spectral Theory of Differential Operators I.W. Knowles and R. T. Lewis reds.} © North-Holland Publishing Company, 1981

A SCHR~DINGER OPERATOR WITH AN OSCILLATING POTENTIAL

Allen Devinatz l Department of Mathematics Northwestern University Evanston, IL 60201 Peter Rej to 2 School of Mathematics University of Minnesota Minneapolis, MN 55455 A relatively large amount of excellent work has been done during the past decade on general spectral and scattering theories for Schrodinger operators with long range oscillating potentials. However, most of these works do not include operators of the form H = -6

+ c sin br + Vex), r

where V is a short range potential. When 17 is radially symmetric, the problem has been successfully dealt with in recent years. On the other hand when V is not radially symmetric only one recent paper deals with these operators, but only for high energy values. In this paper we shall consider the spectral theory for this specific operator and compare our results with the previously mentioned paper.

§l

INTRODUCTION

In this paper we shall consider the problem of the location of the absolutely continuous spectrum of the self-adjoint realization of the Schr~dinger operator (1.1)

H = -6 + c sin br + Vex), r

where V is a short range potential of the type introduced by Agmon [1]; i.e., V is a real valued function in L~ (~n) for which there exists an E > 0 so that ;coc

(1.2)

(1 + Ixl)I+E Vex)

defines, as a multiplication map, a compact operator from the Sobolev space into L2.

~2

Schrodinger operators with oscillating long range potentials have been well studied during the past decade [1-11, 14-17]. In a recent interesting paper, Monique Combescure [4] has studied, among other things, a spectral and scattering theory for operators of the form (j,

(1.3)

H = -6

+ c sin br S r

+ Vex).

However, her results are valid only for a linear combination of (j, and S sufficiently large, and do not include the case a = S = 1. In case V is radially symmetric, the spectral and scattering theory for operators of the form (1.3) has been worked out for essentially all values of (j, ~ 0 and 0 < S 2 1 (see [2] and [17]).

107

108

A. DE VINATZ and P. REJTO

In another recent interesting paper, K. Mochizuki and J. Uchiyama [11] have, among other things, considered a spectral theory for operators of the form (1.1). We shall compare our results with theirs after we state our main theorem. In the theorem which follows, and in the remainder of the paper we shall take (1.4)

-11


0, is in the absolutely continuous spectrum ~ H if

I c I /1 -2v '0 r:;-

(1. 6)

YAO

do ,

0)




(3.21 )

0

a

holds then (.,.) cis an inner-product on H ,as a collection of absolutely cony, y,6 tinuous functions on [a,bJ, with the null element the null function on [a,bJ, i.e. f = 0 in Hy,o if and only if fix) = 0 (x E [a,bJ); (2)

if y

=

cS

1

=2

b

TI

and

J q(x)dx

=

0

(3.22)

a

then (. ,. L. J. is a quasi-inner-product on HJ• J. ; as in [1, section 3J this may "21T ,'z1T "z1T, "z1T be considered as an inner-product on the collection of equivalence classes of functions generated by the null class consisting of all f given by (3.19) for all K E C.

Classical arguments show that in case (1) above Hy,us is complete, [1, section 4J, in the norm derived from the inner-product (. ,.) y,us' i.e., Hy, us is a Hilbert space of point-wise, absolutely continuous functions on [a,bJ. Details of the completion argument are omitted (but see, for example, Coddington and de 5noo [5, p. 158J) except to remark that completion is first proved when r is null on [a,b], and then extended to the general case by means of the unitary (isometric) transx fix) exp[-J r(t)dtJ (x E [a,b]); see Everitt [12, sections formation (Vf)(x) a 4.1 and 5.1]' Similar arguments show that in case (2) HJ J , ' as a collection of equiva;.zlT '~21T lence classes, is complete in the norm derived from the inner-product, and, in this sense, is a Hilbert space. It will be shown below that case (2) is a pathological exception for the general boundary value problem (3.1) and (3.2).

w.N. EVERiTT

132

If from the definition (2.3) of the domain OeM) the domain O,(M) is defined y,u by

oy,u,(M) =

{f

E

OeM) I f(a) =

°

if 'I

= 0, feb) =

°

if 6

= O}

1 then the following relationships may be seen to hold, for all y, 6 E [0, 2 TI],

oy, u,(M) H

'1,6

Hy,o,c AC[a,b] = O(S).

C

(3.23)

There is a need, in general, in case (1) above to reduce the Hilbert space (a,b) in order to take into account the linear manifold G as defined by

r,w

°

(2.33), i.e., the manifold of solutions of Sew] = on [a,b] as discussed in detail in section 2(d) above. This reduction means that, in general, the appropriate Hilbert space for the left-definite boundary value problem (3.1) and (3.2) depends not only on the coefficients p, q and r of M, from the definition (3.16) and (3.17), but also on the coefficients p and W of S. The method used here follows that adopted by Everitt [10, section 2]; see also Oaho and Langer [6]. Suppose case (1) above to hold, i.e., (3.20) and (3.21), and recall the definition (2.33) of G w; define the linear manifold G ,c G by

"v -

p,

Gy,6

{f

=

E

Gp, wi f

E

p,w

Hy,v,L

(3.24)

-

Define the sub-space H,,0,of H'1,6 as the orthogonal complement, see [1, section 7], of G s in H s ' i.e., y~1.)

y~u

H'1,6 We note that H

'1,6

H '1,6

e

{f

Hy,o I (f,g)y,O

E

G y,6

is the largest sub-space of H

,,6

f

E

Hand S[f] = '1,6

=

°

(gEG

,)}.

'1,0

(3.25)

such that

° if and only

if f = 0.

(3.26)

It will be shown in the next section that, under certain additional conditions, the Hilbert space Hy,o,is appropriate for the definition of a self-adjoint operator to represent the left-definite boundary value problem. Right-definite problem For this problem the basic conditions (2.1) and (2.7) hold, together with the specific right-definite conditions (3.13), (3.14) and (3.15); in particular p = andw2-0on [a,b], and"

1

°

1

6E[-2TI'2TI].

In this case the appropriate Hilbert function space is L~(a,b), i.e., the collection of all complex-valued, Lebesque measurable functions f on [a,b] such that

f a

with quasi-inner-product

b

w(x)lf(x)2 Idx


0 (almost all x E [a,bJ) then this null element is the set of all f which are zero almost everywhere on [a,bJ; otherwise the null element will contain

certain non-null functions. The elements of the space L~(a,b) are equivalence classes of functions generated by the null element in the usual manner; see [1, section 3J. With this interpretation (3.28) defines an inner-product for L2 (a,b), and the w usual classical arguments show that this space is complete in the norm derived from the inner-product. In this right-definite case there is no essential requirement to reduce the Hilbert space L2 (a,b) by the linear manifold G ,as in the left-definite case; w o,w in fact it may be seen from case (ii) of section 2(d) that G is the above de2 termined null element of L (a,b). However, if it is desiredOt~ work in a space of w point-wise defined functions as elements, rather than equivalence classes as elements, then L~(a,b) can be reduced to a sub-space L~(a,b) by the definition, using the methcx:l of orthogonal complement, -2

Lw(a,b)

2

=

Lw(a,b)

e Go,w

The null element in L~(a,b) is the null function on [a,bJ. It will be shown in section 5 below that every right-definite boundary value problem can be represented by a self-adjoint operator in the associated Hilbert 2 (a,b) or, equivalently, L2(a,b). function space Lw w . (f)

Translation of the spectral parameter A

In section 2 it is shown that it is always possible to translate the A parameter in the equation (2.11), i.e., M[yJ = AS[yJ on [a,bJ, along the real axis of the complex plane and yet retain the symmetric form of the equation; see (2.22) and (2.23). Left-definite case real p (> q, ~ q, .::..

Suppose that the equation (2.11) is in the left-definite case and consider a translation A = v + T; in the translated equation (2.22) the coefficient 0) is invariant, but the coefficient q is changed to q given in (2.23), i.e. q, + ,w - 2, im[rJ - ,2p2p-l and q T mayor may not sati~fY the condition 0 on [a,b]' Thus translation mayor may not leave invariant the left-definite

134

W.N. EVERITT

property of the equation. It can happen that a left-definite problem is not left-definite under any real translation T f 0; as an example consider the left-definite equation -y" = ii-y'

i.e., p = 1, q rea 1 T f o.

r =w

on

[a,b],

0, P = ton [a,b]; then qT = -

t T2


[1] (. ,I-o) - x[1] (. ,Hcr)~(' :r_o)]b

a

= [x(·,i\+a)¢[1]

"A-a

(·.I-a) -

+ [i (A-a)p (.

xfl] A+a

(.,i\+cr);p(.,I_a)]b a

b

>x (. ,A+a );(. ,I-a)+i {A+a)p {'lx(' ,1.+0 );(. ,I-a )]a

REGULAR DIFFERENTIAL EXl'RhSSlONS AND RELATED OPERATORS

139

(4.15)

The left-hand side of (4.16) is also equal to, on using the differential equation (4.1) , b

J "¢(·,i-o) Slx\",Ho)]

(Uo)

a b

f x(- ,Ha)

- (A-o)

$[(. ,I-o)J

a b

[(Ho) - (A-oJJ

f -;p(. ,I-a) S[x(- ,Ha)J a -

-

+ (A-a) [2ip(-)x(·,Ha)<j>(.,>.-0)]

on using the Green's formula (2.10) for S. [1 J [1 J [ x(·,Ha ) ¢J:-a (',\-a) - x Ha b = 20J

b

(4.17)

a

Hence from (4.16) and (4.17)

- b (·,A+a)<j>(·,A-o)]a

-;j;(',i-a) S[X(',Ho)J - 20[ip(·lx(·,A+a)-;p(.,>:-a)]~_

a

(4.18)

Now

x(a'A+a)<j>&:~ (a,i-a) - x~~; (a,\+a)"¢(a,i-a)

W(x,~)(a,A+a)

=

(4.19 )

since ~(a,\) and ~~lJ(a'A) are real and independent of A. Also from (2.26) b

"¢(b,i-a) = ¢(b,\-o) exp[-J J(A-o) where, for all \

a E

C, b

b

exp[-J J(A) = exp[-J (r a

a

r-

-1

(4. 19a)

2iAPP }J

and from (2.28)

¢Il] (b,I-a) A-a

=

¢~lJ A-a

(b,A-o) exp[-JbJ(A-O). a

Hence, recalling that x(b,\) and xf1](b,A) are real and independent of \, [1]

-

x(b,Ha)¢X_a(b,A-O)

-

[1]

--

xHa(b,Ho)¢(b,A-a)

= (X(b,A-O)¢~~~(b,\-o)

-x~~~(b'A-a)¢(b,A-a)}

x

exp[-fbJ(\-o) a

b

= W(x,¢)(b,A-O) exp[-[ ](\-a) a

= W(x,¢)(a,\-a)

(4.20)

on using (2.21). From (4.18), (4.19) and (4.20) we obtain, with

a

f 0,

140

W.N. EVERiTT

-(20)-1 {W(x,q.)(a,>.+o) - W(x,q.)(a,>.-o)} =

f

b

~(·,I-o)S[X(·,Ho)]

_

_

b

- [ip(·lx(·,ic+o)q.(·,A-O)]a

a

Let 0

+

°to obtain (recall W(x,q.)(a,·) is holomorphic on C) b

-W'(>.) = -[ip(.)x(·,A),¢(.,I)]b + f ,¢(.,I)S[X(·,A)] a a

tlT J),

As in the proof of lemma 4.4 (recall Y, 6 E (0, (x(·,A),q,(·,I))

y,

6

= [x[l](.,A)"¢(.,I)]~

+ Af a

b

(4.21 )

~(.,I)S[X(·,A)]

+ cot o'x(b,ic)"¢(b,I) + cot Y'x(a,A)"¢(a,I)

[xfl](">')"¢(''\)J~ - AW'(A) + cot o'x(b,A)"¢(b,I) + cot Y'x(a,A),¢(a,I) on using (4.21).

From the initial conditions (3.3) this gives

[x(·,A),q.(·,I))y,6

=

-xfl](a,;\.)'¢(a,A) + cot Y·x(a,>.)¢"(a,>.) - AW'(A) - x~lJ(a'A)~(a,>.) + x(a'A)¢~lJ(a'A) -AW' (A) W(>.) - AW' (A)

(A E C).

If An is an eigenvalue of the problem (4.1) and (4.2) then An E Rand W(An) = 0; also if ~n is the eigenfunction then from section 3(b) we have (4.22)

x ( . , An) = kn q. ( . , An) with kn f 0; for all three cases of (4.15) this gives kn (¢(· ,An),cp(' ,An))y,O

(4.23)

-An W' (An)'

=

From this result and lemma 4.3 it now follows that W' (An) f 0 so that all the 0 zeros of W(x,cp) (a ,.) are simple (and real). From (4.23) it also follows that _k- l A n

n

W'(A ) n

>

0

and the normalized eigenfunction can be defined by, for all x n E N, ~ (x) = {-k / (A W' (A ))} 1/2 ¢(X,A ) n

n

n

n

(4.24)

(n E N)

n

E

[a,b] and all (4.25 )

where the positive square root is taken. This last result should be compared with the results in Titchmarsh [22, section 1.9J.

141

RliGULAR DIFITRLNTIAL }iXPI 0 there exists A(E) > a such that

a:

b

J

a

2

b

lal If I ~ E J plf' - rfl a

2 +

A(E)

b

J

a

qlfl

2

(4.31 )

for all f EO Hy, o. Proof.

See [11] and [12] as quoted above.

Proof of lemma 4.8(a)(;;i). In the proof we use K(A) to represent a positive number depending only on A, but not necessarily the same number on each occasion. Consider (·,\;f) with f EO Hy,o ; following the method used in the proof of lemma 4.4 we obtain (recall S[f] EO L(a,b)) -vII (·,A;f)1I 2 0 y,

b

=

im['.l:'

J a

S[f]]

REGULAR DIFFERENTIAL EXPRESSIONS AND RELATED OPERATORS

i.e. ,

Ivlh(.,A;f)1I

2

6:: y,

143

b

IAI suprl(x,O;Tg) = H(X,\;1>(' ,0;Tg)); 1Ji(,,)

(\ E C \

{A :

W.N. EVERITT

152

i.e., for 9

E

O(T), for all x

E

[a,b], and for all ).,

(X,A;g) = A-

l

C\ R

E

{-g(x) + jJ n n

with convergence in the norm of Hy,us again follows that N = N =

(g

E

H

(4.64)

s)

y,u

Also, since Hy,u"has infinite dimension it

00.

Consider now corollary 4.9(b). The proof of this result follows from an identical application of the result in Titchmarsh [22, section 2.13J. Remarks (i) The case when p(x) 0 (x E [a,bJ) is included in the self-adjoint case (4.46); in particular this covers the left-definite case of the symmetric differential equation

M[Y] = ).,wy on [a,b]

(4.65)

with symmetric boundary conditions (4.2), when w is not null on [a,b] but can be of arbitrary sign and can vanish on a set of positive measure within [a,b]. (ii) We have not invoked the theory of compact operators in the proof of the results of this section but this does provide an alternative method to prove theorem 4.9; see the results given in [3, chapter 7J, or, in particular, the account in Taylor [21, section 6.41) which is appropriate to the results considered in this paper. 4.10.

The symmetric case Consider now a return to the general symmetric case of the boundary value problem (4.1) and (4.2) but now without the self-adjoint condition (4.46). It is shown in this section that much less can be proved in this case, in comparison with the self-adjoint case of the previous section; however it can be shown that the problem always has a strictly countable number of eigenvalues, i.e., N = 00, or, equivalently, that the Wronskian W(x,~)(a,·) has an infinity of zeros (all real and simple). It does not seem to be known in this case, however, if the eigenfunctions {>jJn: n E N} span the whole space Hy, us' or even if the projection _ of the {>jJ n : n E N} into the reduced space Hy,us has a linear hull which is dense _ in Hy,us' -

We start by introducing two sub-spaces of the space Hy ,6:

153

RECCL.4R DlFFhRhNTL4L J;X.l'RLSSIONS AND REL/I'lFD OPERAFORS

(i)

Let the sub-space o Hy,o of

Hy,o be defined by Hy,us: f(a) = f(b) = OJ

{f E (4.70) oHy,o noting that the restriction is unnecessary if y = 0 or 0 = 0; if y = 0 = 0 then 'but , otherwise OHy 0 is a strict sub-space of Hy,us; however it is not oH0,0 = H0 0 · difficult to see that Hy,o e 0 H y,o can be at most two dimensional (indeed a basis for this space can be constructed from the two solutions ¢(. ,0) and X(· ,0) of (4.1)); thus we have = co; (4.71) dim oHs y,u

(i i) Let ¢ : [a,bJ + D(M) be defined as the unique solution of the nono homogeneous boundary value problem

M[yJ = S[¢(· ,O)J on [a,bJ

(4.72)

where cp(. ,0) is the solution of (4.1), with tions (3.3). Similarly let Xo: [a,bJ

+

= 0, defined by the initial condi-

A

D(M) be defined as the unique solution of

M[yJ = S[x(· ,O)J on [a,bJ

(4.73)

Standard existence theorems, suitably extended to the case of the genera li zed differential equat ion (4.1), show that ¢o' Xo exist, and since ¢o' Xo E D(M) it follows that
157

REGULAR DIl-'l+RENTI.'l L L'(PRESSIONS AND RELATED OPERATORS

or are indeterminate due to technical difficulties; it seems to be an open question as to whether or not {~n} is complete in o Hy, 6' in Hy, 8' or in Hy, o' (iii) Another open question would seem to concern the order of W(x,~)(a,·) as an integral (entire) function on C; all the examples point to the order being 1 if p is not null on [a,bJ, and ~ if p is null on [a,bJ; if this is the case then no information concerning the number N of eigenvalues can be obtained from the theory of integral functions when p is not null on [a,bJ. (iv) The results of this chapter are largely due to appropriate applications of the methods of Titchmarsh [22J; it would be of interest to know if an operatortheoretic proof of theorem 4.10 and corollary 4.10(a) can be given. 4.11.

The left-definite case-examples.

We discuss a number of examples to illustrate the results given earlier in this section. Example 1. Let a also y = 8 = 0, i.e.,

0 and b = 2n, and p = 1, q

-y"= iAY' on [0,2nJ

and

y(O)

w = r = 0, Y(2n)

=

=

p

= ~ on [0,2n];

0;

note that case 1 of section 3 holds, in fact (3.20) is satisfied. This is an example of the self-adjoint case considered in section 4.9. {I} and it may be seen that G = {OJ 0,0 H0,0 = 0 H0,0 = {f E AC[O,2rrJ: frO) = f(2n) = 0, f' E L2 (0,2rr)} with 2n

(f,g)O,O

J

f'

Here

gr.

o

A calculation shows that ~(X,A) =

1

1 - e iA(2rr-x) iA

and (Ic

E

C).

We note that W(x,~)(· ,A) is an integral (entire) function of order 1 with zeros at the points An = n (n = ~l ,~2, ... ); note also that 0 is not an eigenvalue (see lemma 4.2). The normalized eigenfunctions of this problem are, for n = ~1,~2, ... , ~n(x) = n- I (2rrr l / 2 (1 - e- inx ) (x E [0, 2rrJ). It

may be seen directly that

1 2n . . .1,) = -- J e- 1mx e 1nx dx = 6 . m'o/n 0,0 2IT 0 m,n Either from theorem 4.9, or directly, it may be seen that the set

(~

{~n}

is complete

WoN. EVERITT

158

in H Note that, in the sense of the inner-product in H0,0 , the function x on 0,0 [0,211] is orthogonal to all the {~n} in H0,0 ; however x f/; H0, 0 since this function does not vanish at the end-point 211. It is possible to calculate the form of the operator T of section 4.9 in this case; we find X 211 (R f)(x) = <Jl(x,O;f) " {(x - 211) J tif' (t)dt + x J (t - 211 )if' (t)dt} o 11 0 X

i-

for all x E [0,211], and from this a calculation shows OtT)

=

{g: [0,211] ...,. C 9 and g' EAC[0,211]' gil n 2 (0,211), g(O) 1

g(211)

OJ

and (x E [0,211]).

(Tg)(x) " i (9' (x) - g' (0))

It has to be regarded as exceptional that we can calculate T explicitly. Note that T is obtained from a differential expression of the first-order; formally this is in line with writing T = (M-1S)-1 even though no definite meaning can be given to the right-hand side. Example 2. 0,6

y

=

~11'

Let a " 0 and b " 211, and p

=

1, q

i.e., -y" " iAY' on [0,211] and Y(O)

=

21 on [0,2 11 ]; also y'(211) + ~ iW(211) = O.

w = 0,

= 0,

p "

This is an example of the symmetric, non-self-adjoint case of section 4.10; here again Gp ,W = {l} but G0, 1"ZiT = {OJ so that H0 , 1'ZTI = H0 ,''2TT 1 with 2 H ,1 = {f E AC[0,211] 1 f(O) = 0, f' E L (0,211)} O 211 and H = {fEH If(211) = OJ; o 0, ~iT 0,'211 also 211 (f,g)o , "11 = J f' go. 2 0

-

1

I

Here cp(x,A)

- e -iAx

and W(X,CP)(O,A) =

t

(1 + e

iA211

)

which again is an integral function of order 1. The eigenvalues are given explicitly by An = n + ~ (n = O,~l ,~2,···) which are strictly countable in number, see corollary 4.10(a); note again, see lemma 4.2, that 0 is not an eigenvalue. The normalized eigenfunctions are given by (x

o

H

A straightforward argument shows that the set but al so in the whole space H 1 = H"

O,~TI

O,YzTI

O,~TI

{~n}

E

[0,211]).

is complete not only in

159

RDCL'LAR DIFFliRliNTIAL LX1'RESSIONS AND Rf:L41'liD OPERATORS

We have also

f

~(x,O;f)

x

f

2TI

(x E [0,2 TI ]). f' (t)dt o x o and this follows directly from Since y = 0 we have already ~(O,O;f) However the above; thus the reduction (f,x) I = o is automatically satisfied. o 0, ~2TI 2 reduction (f,~) I = 0 is necessary; it may be seen that ~ (x) = x /(2i) o o 0 ,'2TI (x E [0,2TI]) and so 2TI 2TI if f(t)dt; f tf' (t)dt (f, ~o)o , l 0 (almost all x EO [a,b]); see again the remark at the end of section 3(d). The identification of the operators in the two definitions is given by showing, with w > 0 on [a,b], that {R } A -1 is the resolvent family of the operator generated by w M. Proof. The proof of theorem 5.9 follows the same 1 ines as for the proof of theorem 4.9; the details are omitted. 0 Corollaries 4.9(a) and 4.9(b) extend as given to the right-definite case replaced by L2 and i 2. 0 with Hand H y,o y,6 w W

RcGUL4R DlFFl'.RbVTIA I. l:XPIU:SS[ONS ,1ND J(cLUl:D OPIOR.-Jl'ORS

5.10.

165

The symmetric case.

As mentioned in section 5.8 there is no need to consider this case separately for the right-definite boundary value problems. 6.

GENERAL REMARKS

We have considered certain of the spectral properties of the second-order, linear, symmetric differential equation M[yJ = AS[y] on [a,b], with associated, separated boundary conditions. In the right-definite case (see section 5), and in the left-definite, selfadjoint case (see section 4) it is shown that the boundary value problem can be represented by a self-adjoint operator in a suitably chosen Hilbert function space. The methods employed are those of classical complex function theory and operator theoretic properties in Hilbert space theory. In the left-definite, non-self-adjoint case no satisfactory operator representation of the boundary value problem seems possible but the existence of an infinity of eigenvalues is obtained by an adaption of the classical methods of Titchmarsh. We list here a number of open problems: (i) What can be said of the order of W (the generalized Wronskian defined in (3.4)) as an integral (entire) function on the complex plane? (ii) What can be said of the zeros of Wwhen the general boundary value problem of section 3(b) is neither left-definite nor right-definite? (iii) Under what conditions on the coefficients of the differential equation can asymptotic expansions for large values of A be obtained; are such expansions uniformly valid on [a,bJ? (iv) In the left-definite, non-self-adjoint case of section 5 do the eigenfunctions form a complete orthonormal set in H, or Hy,u~? y, u (v) Does the analysis of Daho and Langer [6 and 7J extend to the leftdefinite case when one or both of the boundary condition parameters y and 0 lies in the negative interval [- ±rr,O]?

166

W.N. EVERITT

References [lJ

Akhiezer, N. I. and Glazman, I. M. Theory of linear operators in Hilbert space: Volume 1 (Pitman; London and Scottish Academic Press; Edinburgh, 1981; translated from the third Russian edition).

[2J

Atkinson, F. V., Everitt, W. N. and Ong, K. S., On the m-coefficient of Weyl for a differential equation with an indefinite weight function, Proc. London Math. Soc. (3) 29(1974), 368-384.

[3]

Coddington, E. A. and Levinson, N., Theory of ordinary differential equations (McGraw-Hill; New York, 1955).

[4]

Coddington, E. A. and deSnoo, H. S., Regular boundary value problems associated with pairs of ordinary differential operators, Proceedings of Equadiff 78 (International Conference on Ordinary Differential Equations and Functional Equations, Florence, Italy 1978).

[5]

Coddington, E. A. and deSnoo, H. S., Differential subspaces associated with pairs of ordinary differential expressions, J. of Diff. Equations 35 (1980), 129-182.

[6J

Daho, K. and Langer, H., Some remarks on a paper by W. N. Everitt, Royal Soc. of Edinburgh (A) 78 (1977), 71-79.

[7]

Daho, K. and Langer, H., Sturm-Liouville operators with an indefinite weight function, Proc. Royal Soc. of Edinburgh 78(1977), l61-19l.

[8]

Dunford, N. and Schwartz, J. T., Linear operators: Part II (Interscience; New York, 1966).

[9J

Eastham, M. S. P., Theory of ordinary differential equations (Van Nostrand Reinhold; London, 1970).

Proc.

[lOJ Everitt, W. N., Some remarks on a differential expression with an indefinite weight function, Mathematical Studies 13 (1974), 13-28 (North Holland; Amsterdam, 1974; edited by E. M. de Jaeger). [11J Everitt, W. N., An integral inequality associated with an application to ordinary differential operators, Proc. Royal Soc. Edinburgh (A) 80 (1978), 35-44. [12J Everitt, W. N., On the transformation theory of ordinary second-order linear symmetric differential equations, Report DE 81: 1, Department of Nathematics, University of Dundee. [13] Everitt, W. N. and Race, David, On necessary and sufficient conditions for the existence of Caratheodory solutions of ordinary differential equations, Quaestiones Mathematicae 2 (1978), 507-512. [14J Everitt, W. N. and Zettl, Anton, Generalized symmetric ordinary differential expressions I: the general theory. Nieuw Archief voor Wiskunde (3) XXVII (1979),363-397. [15J Kamke, E., Differentialgleichungen: Losungsmethoden und Losungen (Chelsea; New York, 1971; reprinted from the third edition, Leipzig, 1944). [16J Naimark, 1968) .

M. A., Linear differential operators: Part II, (Ungar; New York,

REGULAR DIFFERENTiAL EXPRnSS]ONS AND RFL1TrD OPERATORS

167

[17J Niessen, H. D. and Schneider, A., Spectral theory for left-definite systems of differential equations: I and II, Mathematical Studies 13 (1974), 29-44 and 45-56 (North Holland; Amsterdam, 1974; edited by E. [q. de Jaeger). [18J Pleijel, Ake, A positive symmetric ordinary differential operator combined with one of lower order, Mathematical Studies 13 (1974),1-12 (North Holland; Amsterdam, 1974; edited by E. M. de Jaeger). [19J Pleijel, Ake, Generalized Weyl circles, Lecture Notes in Mathematics 415 (1974), 211-226 (Springer-Verlag; Heidelberg, 1974; edited by 1. M. Michael and B. D. Sleeman.) [20J Riesz, F. and Sz.-Nagy, B., Functional analysis (Ungar; New York, 1955). [21] Taylor, A. E., Introduction to functional analysis (Wiley; New York, 1958). [22J Titchmarsh, E. C., Eigenfunction expansions I (Oxford University Press, 1962)

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Spectral Theory of Differential Operators I. W. Knowles and R. T. Lewis leds.) © North-Holland Publishing Company, 1981

AN EIGENFUNCTION EXPANSION ASSOCIATED WITH A TWO-PARAMETER SYSTEM OF DIFFERENTIAL EQUATIONS Melvin Faierman Department of Mathematics University of the Witwatersrand Johannesburg South Africa Dedicated to Professor F.V. Atkinson on his 65th birthday

In this work techniques from the theory of partial differential equations are used to prove the uniform convergence of the eigenfunction expansion associated with a left definite twoparameter system of ordinary differential equations. INTRODUCTION We consider the simultaneous two-parameter systems (p] ( x] )y; ) + (A] A] (x] ) - A2 B] (x] ) - q] (x 1 ) ) y] 0, 0 R} we notice that E(R) tends to 0 when R + +00. Let aq be an integrodifferential form, continuous and coercive on V~([J):

(2)

a (u,v) = (a+q)(u,v)

=

q

f ( T Q

I a T.::.m

a sex) Oau(x) OSv(x) + q(x)u(x)VTXTT dx a

I si.::m for (u,v) (3)

E

VO((l) q

x

VO([J).

We suppose that:

q

a

as

=

-a-

Sa

E

CO(i'l) and that:

Let us denote by AO the positive selfadjoint operator, unbounded in q associated by the Lax-Milgram theorem, to the variational problem (V~([J), L2([J), aq).

2

L (n),

175

DISTRIHUTION OF UGh",!' lLUES OF SCHROEDINGJ:R OPhRATORS

We deduce from the Proposition 1 that AO has a discrete spectrum consisting q of isolated eigenvalues: s.

---r

J j

-+

+00

+00

(each eigenvalue is repeated according multiplicity). We study the asymptotics of the number of eigenvalues less than s: N(s,A o ,Q) = N(s,Vo(Q),a ) = card{j E JIl / sJ. < s} when s q q q We give now some assumptions concerning A and Q.

->

+00.

q

There exist two positive numbers EO and s· such that be extended to ~ = {x ERn / dist(x,Q) < EO} and (4)

\I EEl 0, E [, \Is > s· o \I(x,y) E

Qx n

\Ill .':. lls q(x)

=




Let (V,H,a) and (V,H,b) be two variational problems such that: 0, 3c

>

0, Ifu

E

V Ia ( u , u) - b (u , u) I ~ Ea (u , u) +

q uII ~

177

DISTRIlJUTION OF EIGENV IILt'ES OF SCHROEDINGfiR OPERATORS

N[O-ds - C, V, b) :5- N(s, V, a) :5- N(s(l+d + c, V, b). We suppose now that Q is an open set in R n and Hm(Q) and Hm(Q) are the o usual Sobolev spaces on Q. Let a be an integrodifferential form, hermitian, continuous and coercive on Hm(Q). We denote by Al Crespo AO] the realization of the variational problem (Hm(Q), L2(,,), a) Crespo (H~("), L2(Q) ,a)]. ~~Q~Q~~!~Q~_~:

Suppose that Ql and Q2 are two disjoint open sets in Rnsuch that: = "1 U Q2; then, the following holds: l l N(s,Ao ,Ql) + N(s,Ao '''2) :5- N(s,Ao ,Q) :5- N(s,A ,,,) :5- N(s,A ,Ql) + N(s,A l '''2)' where N(s,Ao,n) = N(S,H~(,,),a) and N(s,A l ",) = N(s,Hm(Q),a). Remark: This result can extend to other spaces and, in particular, proposition 4 holds for the spaces V~(Q) introduced above (i = 0 or 1 correspond to different boundary conditions). IV - A FIRST ESTIMATE Let us write:

f(s,q)

=

f

\l(x) (s_q(x))n/2m dx; Q

s

the following estimate holds. THEOREM 2:

Proof:

There exist two positive numbers c' and c" such that:

c' s n/2m [Q s J -< f(s,q) c " S n/2m [ Q s ] We deduce from the coerciveness of aq that:

lat=m

\;Js .:: s" •

r;2a

hence: \l(x) :5- cp-n/2(x), and we obtain the upper bound. To obtain the lower bound, we use hypotheses (3) and (6). We can write: f(s,q) .::

f

\l(x) (s_Q(x))n/2m dx .:: c[Qs] sn/2m.

ns/ 2 V - ESTIMATES FOR AN OPERATOR WITH CONSTANT COEFFICIENTS ON A CUBE Let A~ be the operator defined on Q~ associated with the hermitian form: (10)

a (u,v) I;

=

f Q

1

\aS <m <m

aaB(X~) Oau(x) OSv(x) dx ;

I; we denote by A~ the leading part of AI; associated with the form following results: ~~gEg~!I!g~_~:

a~.

There exist two positive numbers 60 and Y4 such that:

We have the

178

]. FLECKINGER

'tI Ii

'tI s .:: S", 'tI ~ E I, 'tI n .2 11S' ::'IT t; =

0,

2

(10)

2

with qEL I 1 ~,oo), the basic expansion theory has been given , oc by the author in [6], and the Weyl-Titchmarsh m-coefficient in the limit point case is again uniquely characterized by the requirement, (11 )

where

{ 0 differs by two from the number of L2 solutions for IA < O. REFERENCES [lJ [2J [3J [4J [5J [6J [7J

Gilbert, R.C., Asymptotic formulas for solutions of a singular linear ordinary differential equation, Proc. Roy. Soc. Edinburgh. Sect. A 81 (1978), 57-70. Gilbert, R.C., A class of symmetric ordinary differential operators whose deficiency numbers differ by an integer, Proc. Roy. Soc. Edinburgh. Sect. A 82 (1978), 117-134. Gilbert, R.C., Integrable-square solutions of a singular ordinary differential equation. to be published by Proc. Roy. Soc. Edinburgh. Sect. A. Gilbert, R.C., Shearing transformation ofa linear system at an irregular singular point, to be published by Math. Proc. Cambridge Philos. Soc. Kogan, V.I. and Rofe-Beketov, F.S., On the question of the deficiency indices of differential operators with complex coefficients, Proc. Roy. Soc. Edinburgh. Sect. A 72 (1975), 281-298. Orlov, S.A., On the deficiency index of linear differential operators, Ookl. Akad. Nauk SSSR 92 (1953), 483-486. Warsow. W., Asymptotic expansions for ordinary differential equations (Interscience, New York, 1965).

Spectral Theorv of Differential Operators I. W. Knowles and R. T. Lewis leds.) © North·Holland Publishing Company, 1981

HIGHER DIMENSIONAL SPECTRAL FACTORIZATION WITH APPLICATIONS TO DIGITAL FILTERING R. Kent Goodrich Karl E. Gustafson University of Colorado Boulder, Colorado, USA

A key tool in the theory of digital filtering in one dimension is a certain general spectral factorization. The lack of such factorization has been a major impediment in the development of a digital filtering theory in higher dimensions. We give here a general method for such factorization in any number of dimensions.

INTRODUCTION AND BACKGROUND Two and three dimensional filters are currently under much investigation in the electrical engineering community and are central to many array processing applications. In the one dimensional theory one employs a general factorization of a certain spectral density associated with the process under consideration into the product of an inner function and an outer function. Among all filters which produce the same gain at each frequency, the outer function corresponds to the filter producing that gain with the minimum group and phase delays. Outer functions have no zeros in the upper half plane. Thus all such zeros in the Hardy function being factored have been absorbed into the inner function. The latter is essentially and in many cases a Blaschke product. Because the zeros of functions of more than one complex variable are generally continua, there has been difficulty in extending the filtering theory to more than one dimension. Our method is neW and apparently the first general inner-outer spectral factorization in higher dimensions. Its abstractness, coming from a functional analytic approach and from considerations of stochastic processes in quantum mechanics, has not as yet been tested as to direct applicability to filtering problems. For the moment, it may be viewed as the beginning of a new theory of inner and outer functions in higher dimensions. It also has important implications in higher dimensional approximation theory. We hope to show in this paper its possible implications to digital filtering in higher dimensions. Further details of the analysis and full proofs of a number of the results given here may be found in a paper to appear [1]. A preliminary announcement of some of these results was given in [2], where the emphasis was on the relationship to higher dimensional purely nondeterministic stochastic processes. It should be stressed here that the higher dimensionality is in the parameter variable, and not in the vector valued random variable, which has been and usually can be generalized from the one dimensional scalar range to finite, infinite, and matrix valued ranges. Some further details, especially as to the relations to regular representations of arbitrary groups and to support questions for generating cyclic vectors, may be found in [3]. There also the connection to fundamental approximation problems is emphasized. In this present paper we wish to describe somewhat briefly the results given in [1], also [2] and [3], and moreover to attempt to place them in the context of filtering theory, where their eventual implementation may be of significant practical, beyond conceptual, value. Their connection to the spectral theory of differential operators, the subject of this conference, is threefold. First, as

199

200

R. KENT GOODRICH and KARL GUSTAFSON

is well known, modeling physical systems subject to random inputs yields solutions of ordinary differential equations in terms of realizable convolution filters in a wide variety of situations, such as in the theory and application of Kalman filters. Second, as was established in [4], all square integrable white noise processes are unitarily equivalent to quantum mechanical momentum evolutions. In particular, such evolutions are generated by first order partial differential operators with absolutely continuous spectra the whole real line and with an additional spectral requirement on the spectral density that corresonds to the optimal gain property mentioned above (this will be explained below). Thirdly, these questions can be posed in terms of boundary value problems for the Laplacian on the upper half plane, or, in the higher dimensional cases, the Laplacian in half spaces, quadrants, and other configurations. DIGITAL FILTERING AND FACTORIZATION Two excellent references for these topics are [5] for the filtering theory and [6) for the function theory. Much of what we say here may be found therein, although we will take a slightly different point of view here, stressing the most elementary connections between the filtering theory and the function theory, from the point of view of spectral theory, in order to make the connections to our approach in [1].

Given an input process

X

or

, parametrized here by a one dimensional parameter

t

,which may be time or space and which may be discrete or continuous, a

L : X + Y t t

linear filter is a linear operator X

L2

or both, whenever need be, without further specification thereof.

L1

t

We will imagine all processes as in

t

The resulting transformed process

Y

t

on the space spanned by the

is called the output process.

filter is required to have the "time invariance property"

The

L

Because of this latter property one immediately sees a connection to group representation, which is one facet of our approach in [1). Moreover, for higher dimensional filtering applications the parameter t should be repla ced by a g for a general group, and in particular by v for a two or three dimensional space variable in the group Often

X

t

Y

t

LX

t

B(A)

n =

2,3

t

r

=

fOo.)

V

eiAtf (A)dA = fO 0

_00

has a corresponding representation Y

where

n

has the representation in terms of its spectral density X

Then

R

t

LX

=

t

i (ooe At B( A)f ( A)d lO

V

(BfO)

is called the Transfer Function of the fi Iter

IB( I

i

L

Writing

B(A)

A) e B( X), expresses the transfer function in terms of in polar form, B( A) = A (it may be a loss) and the the "gain" iB(A)1 i t produces at each frequency "phase shift B(~) • Because usually only real processes are considered, so

201

HiGHER DIMENSI01VAL SPEC1RAL l'ACJ"ORIZA'110N

that

X t

and

Y

will be real valued, one has

t

fOe A)

an even function.

An important class of linear filters are the convolution filters 00

Y

The kernel

k

t

= LX

00

(ooX(t-S)k(s)ds = J_ook(t-s)X(S)ds

t

is called the impulse response function. V

(Bf 0)

= k

*

Since

X

by the convolution theorem we have (BfO)( A) But

A

fOe A) = X (A)

=

A A

(K X )( A)

so we see that the transfer function

B( A)

transform of the impulse response function k B(A) = kA(A) important physical restriction (causality) is that k(s) = 0

for

s




(SKKKW)

The latter condition is, as we have indicated, sometimes called the Szego-Kolmogorov-Wiener-Krein spectral condition. Apparently and probably Krylov should be added as he arrived at a similar condition, although in a different context, in [7]. The factorization now comes about as follows. Among the impulse response functions k(s) , also sometimes called the filter by abuse of notation, which produce the same gain IB( A)I ,there is an optimal one kO(s) which is called the minimal delay filter or minimal delay impulse response. from any causal k( s) that produced the desired gain IB( A)I

= B( A)

a function in the Hardy space B( A)

2 H +

This is obtained by factoring k A ( A)

according to:

g( A)1jJ( A)

where 1jJ is an outer function and g is an inner function. Outer functions are characterized by their having no zeros in the upper half plane and by the fact that their absolute values satisfy a Jensen's type equality;

where the right hand side means the Poisson Integral from above 1jJ(0+) on the upper half plane. on the upper half plane and

P

K

of the boundary values

Inner functions satisfy

Ig(A)1

i

1

202

R. KENT GOODRICH and KARL GUSTAFSON

Ig(O+)1 =

a.e.

~

Outer functions are also characterized as all functions

--

V

in

2 H+

such that

2

sp{~ (>--s) Is ~ O} = L (0,00)

Note that

~V

of the given

2

is the inverse Fourier transform of the L (_00,00) ~

boundary values

in

FACTORIZATIONS IN HIGHER DIMENSIONS For the sake of simplicity we state our results for n = 2 parameter'dimensions. Analogous results hold for all n > 1 under suitable modifications. It is useful, both conceptually and practically, to now think of the parameter space, e.g., R2 the Euclidean plane, in a spatial sense rather than in a time sense. This corresponds naturally to studying approximation and digital filtering problems in several variables. There is a great deal of recent interest in signal processing and elsewhere in two and higher dimensional filtering problems. We cannot do justice to the wide and rapidly increasing literature on these problems and applications. As a sample see [8] and the references therein. As stated in [8] and elsewhere, the lack of a general factorization method has been a major obstacle to theory and application in higher dimensions. Remember that, as indicated above, the inner function must, among other duties, remove unwanted zeros. Remember also, as also mentioned above, the zeros of analytic functions of more than one variable may have a very complicated structure. Defini tion.

A function

~

in

2 L (R2)

is an outer function i f

Note the similarity of our definition to the one dimension characterization stated immediately above this section. Let R2

Let

v + U v

be a continuous unitary representation in a Hilbert space

We suppose

U

has a cyclic vector

, and

~O

, i . e. ,

be the projections of y ~

and sp{U(x,y)(o)ly Denote the range of any projection

P

by

H

~ d

R(P)

H

d

onto, respectively:

of

203

HIGHER DIMENSIONAL SPECTRAL FACTORIZATION

Definition:

v + U ( CPO) v

The mapping

is a regular process provided that:

for all

(s,t)

in

and

ns R(E s ) Theorem.

(Representation)

cyclic vector L2(R2)

CPo

Let

{oJ = nR(Ft) t v + U ( CPO) v

Then there exists a unitary mapping

V

of

H

with

H

onto

such that

VU V-I v

where

be a regular process on

R

v

R

=

v

is the regular representation of

R2

Moreover

is an outer function.

vie remark that the regular representation of

R2

is given on

2 L (R2)

by

(R/)(w) = few-v) The proof of the theorem may be found in [1]. The essential ingredient is the Stone-von Neumann Theorem. One uses the identities

and U(X,y)FtU(_X,_y) = F t +y The corresponding representations of W = JeixsdFx s

R1

given by

V t = JeiytdFy

and

satisfy the imprimitivity commutation relations

U

V U

(x,y) t

(-x,-y)

= e-ityV

and U

WU = e-is~ (x,y) s (-x,-y) s

The projection valued measure

p

corresponding (by Stone's Theorem) to

turns out to be quasi -invariant and hence


l

- A)-lfl>1

I jOOeY/Zfz(y)dY

o

J

g(y,(;A)e V2 f (()d(1 l

C16a)

0

(16b) by Schwarz's inequality where

r'e °

Y

IfCy)

sup (II (L

fEV

l

2 1

dy

- A) -lfll/II£II).

(16c) (16d)

IS0M H. HERRON

216

What is needed next is that g(y,~;A) is bounded in A. From (13b), (14a) the only singularities of gCy,~;A) in A can occur where rCA) = O. Now lim gCy,I;;A) = r (A )-+0 [-\y - 1;\ + (y + 1;)]/2, which is finite. Thus condition (i) holds. It follows that Op(L ) is empty since g(y,I;;A) has no poles. Hence O(L l ) = crc(L l ) = oe(L l ). l We take r = cre(L ) = {A € !RIA ~ This half-line is given parametrically by l

i}.

2

(17) 1/4, The verification of condition (ii) follows with the identification; if AO € r, then from (7), AO = w~ + 1/4 for some nonnegative wOo Next take lim+ rCA) = r(A~) = - iwO and lim_ rCA) = r(A~) = iwO in (13b). Thus the limits A-+AO r-+AO exist since the terms involving rCA) are bounded and the conditions on fl and f2 ensure the convergence of the integrals. A = w +

Condition (iii) is the most involved. Let the arclength s = A so that ds = dA = 2wdw. It tallows from (13b) and the last paragraph that

j( IR+(A,fl ,f2) - R-(A,fl ,f2) Ids r

=

fo f

=

co

00

I

f'2CYTdy

0


-

2

denote this by y E LA'

219

bole -+

a

y* AY


n.

It has been shown by Kogan and Rofe-Beketov [11]

that dim SeA) is constant in the upper and lower half-planes. LEMMA 1.1.

16

(1. 3)

hold;."

.the.n dim S (A)

dim SeA)

n

SOfl.

1m A 1

o.

221

TITCHAIARSH-IVEYL THEOR Y

PROOF.

that ~*(a) J di~ts

Sin~e J is non-singu-

If not, then for some A, dim SeA) + dim sCi) > 2n.

lar, then dim J S (i)

1" (a)

=

Hen~e there is a ~ E S (A) and a ~ E S (i) su~h

dim S (i) .

:f o.

A differentiation shows

;* J;

is constant; this contra-

(1.3) and proves the lemma.

A differentiation establishes that if ~ is a solution of (1.1), then (1. 4)

existen~e

To prove the

~lassi~al

of the

meA) function, a regular boundary value

problem is associated with the differential operator.

We follow the same method

and associate with (1.1) the eigenvalue problem (1.5)

o where the n

n matrices aI' a , 13 , 13 satisfy: 2 1 2

x

(1. 6)

The eigenvalue problem (1.5) may be put in the parametric form of the text of Atkinson, i. e. ,

J~' where

M~'JM =

N'~JN

(AA + B)~,

~(a)

and Mu

0 implies u

Nu

=

M

= (

~

a --a

=

,', 2

*

),

Ya'

where J

Green's matrix

Y'a

= (AA

+

whi~h

B)Ya ,

Nv

=

for some v f 0

c; ~ ).

0, by choosing, N

l

The above problem is symmetric and has no asso~iated

~(b)

Mv,

-i3'~

1

~omplex

eigenvalues.

may be constructed.

Thus there is an

Take as a fundamental matrix

and

Ya (a) The conditions (1.6) ensure that Ea is non-singular. loss of generality that

..,.

We partition the matrix Y into n a

x

n blocks by

We further assume without

222

DON B. lIINH)N and K. SHAlt'

Y (x,A) a

~(X'A»)

__ (S(X,A) .. S(x,A)

¢ (x, A)

and also use the notation

3(X,A)

=

SCX,A») ( SCx,A) A

,

~(x,A)

(¢(X,A») =

A

Ijl(X,A)

These are the matrix analogues of the scalar functions (cf.

e, ¢

used by Titchmarsh

[6]).

The Green's matrix of (1.5) [1, p. 265] is given in terms of a characteristic function F = FMN(b,A) [1, p. 269].

In our notation, F satisfies (for A not an

eigenvalue) Ea

l

(FJ

1 + -2

I)E

(1. 7)

ex

Some calculation reduces (1.7) to (1. 8) where (1. 9)

A property of F that is crucial to our development is [1, p. 289]:

A slight modification of the proof of [1] shows F is uniformly bounded in band A for A restricted to a compact set containing no real numbers. Following now the scalar case we compute a 2n x n matrix solution ~b of (1.1) of the form ->-

(8

+

->-

(1.10)

¢ C

->-

which satisfies the right-hand boundary condition of (1.5), i.e., [Sl,S2]'I'b(b) =0. Substitution of (1.10) into this boundary condition yields that C = Ma(b,A) where Ma(b,A) is given by (1.9). LEMMA 1. 3.

We now investigate the behavior of Ma(b,A) as b

->-

b 1,.

223

TJTCIIM.JRSIl-II'EYL THEORY

PROOF.

From the relations

[ 8~] 0,

n,

=

-8~

we conclude that for some matrix

r,

->

~b(b)

[8 ,-8 1*r. Z l

A short calculation now

yields the conclusion. -+

Replacing y by

-+ ~b

in (1.4) yields (1.11)

16

THEOREM 1. (i)

dim SeA) = dim SeA) = n bOJL 1m A .; 0, then bO!L all A wilh Im(A) f 0,

M (A) =: lim;, Ma (b, A) eJe-ud;, clf1d ;;, ~ndependent a b+b Ma (A) ;;, analyulC and IW6 !Lank n;

0

b

8

1

and 8 ; further Z

(ii) (iii)

(iv) PROOF.

the m~x M CA) - M'~(A)/Im A ;;, pMili.ve deMnLte; M (A)

a

a M'~(i).

a

a

From (1.8) and Lemma 1.2 we conclude that for b

-+ b* as n + 00, {Ma(bn,A)} n C as n -+ 00 through a subsel+ a n ""* quence. Letting b = b in (1.11) and defining IjIl = Y [I ,C*l'~, we conclude that n a n 1 the columns of 11 are in L~. The matrix In in the definition of ~l implies the

has a convergent subsequence.

Suppose M (b ,A)

-+

-+

columns of IjIl are linearly independent; hence they are a basis of SeA).

If C

is

2

the sequential limit of another sequence {Ma(dn,A)}, then similar reasoning may

- Ya[I ,C;11,. Since the columns of both 11 and n Z we have for some n x n matrix C , be applied t01

>P2

span sO),

3

hence C = In and C = C . This establishes the existence of the limit. Similar l 2 3 reasoning shows the limit is independent of 8 and 8 . The analyticity of Ma(A) Z 1 follows from the uniform boundedness properties of F. The relation (1.11) yields a proof of (iii).

The relation (iii) implies the rank of Ma(A) is n.

To establish (iv), note that the choices (Sl

I

n

)

in

(1.9) yield by (i), M

a

CA)

lim hp (b ,A) -1 b+b;'

e (b, A) }

lim { - $ (b , A) -1 b-+b*

e (b, A) }.

(LIZ)

DON B. HINTON and K. SHAW

224

A differentiation shows y1'(b,~) JY (b,I) B y1'(a,~) JY (a,I) a

a

a

a

J.

Reversing the

order of products gives that

The upper left-hand corner of this equation is

o

= e(b,I) ¢* (b,A) - ¢(b,i)

e;'

(b,A).

This relation and (1.12) completes the proof of (iv).

PART II.

SPECTRAL THEORY

The inner product defined by ->-->-

generates only a seminorm

f

b*+ ->f>~(t) A(t) get) dt a

II ->-f IIA =

-)- 1> 11/2 ,unless A(x) is invertible. [ 0, in the form (1.1) we take ([1, p. 253])

For the Dirac systems of [12] we note that A(x) = I

2n

.

To allow for these cases

we will assume henceforth that A(x) has the form A(x) __ [A10 (x) where A1 is r x r and invertible, r

2n.

Letting

{E)~Tg E L!}, we note that Er L! is a Hilbert space under the inner

2

and Er LA product
t,

K by

and define an operator

f

b'~

-+

K(X,t,A) g (t) dt

(2.3)

a

for

gE

L2 and 1m (A) A

+ O.

We note that (2.2) can be written, for 1m (\)

if

a

K(X,t,A)

if

(X,A) [ 0 I n

If we let

y=

(t,);');',

0

as

x < t,

a (2.4)

~ * I ] -+ Y (t,A) , M (\) a a

(x, \) [ 0

a

o ] if M (A) a

+ 0,

x > t.

K(X,A,g) and use (2.4), then direct differentiation gives (2.5)

-+

since Y is a fundamental solution matrix. By an identity in [1, p. 269], the a right side of (2.5) reduces to A(x) g (x), or in other words T = g. These steps

y

are permissable even if \

=

AO is real, provided M(A) is analytic at AO.

However,

we have to show that (2.3) is actually defined for real \0; i.e., we need to establish that the columns of ~(X,AO)

8(X,A ) + ~(X,AO) Ma(AO) belong to L!. O

For

this we rely on the identity fb a

1 '

~;'(t,A) A(t) lJI(t,A)dt

which is the limiting case of (1.11). analytic at AO' recall that ~

1m Ma(A) 1m (A) f 0,

1m A Put A

e + $ Ma(\)

=

(2.6)

AO + iv in (2.6), suppose Ma(A) is

and let v

-+

O.

The right side of

(2.6) approaches M' (A )' and so an appeal to the Lebesgue convergence theorem O -)b~'( -)lJI(t,A ) A(t) lJI(t,AO)dt < ro Standard operator-theoretic arguments may O now be used to identify (2.3) with the resolvent operator. Hence AO E peT) whenyields J a

ever AO is a regular point of Ma(A).

227

'111'CHM.. } RSH-It'rYL THEOR)'

The basis for the other direction in (i) is the identity, valid for 1m (A) M CA)

(A - i) f

M (i) C!

C!

+

(A

+ 1) f b""'

2

b . .'~

.-+....

-1-

,¥"(t,i) A(t) 'I'(t,i)dt

a

-

+ 0,

+

K(t,A, '!' (',i»

(2.7)

* A(t) --+-'¥(t,i)dt,

a

whose derivation may be found in [10].

If we start with AO E peT) then a separate

argument, based on the fact that K(t,A,') is known to be the resolvent operator of T for 1m (A)

+ 0,

establishes that the right side of (2.7) is analytic in a

neighborhood of AO (see [10]).

Thus (2.7) gives the analytic continuation of

Ma(A) to AO' Concerning (ii), to say that Ma(A) has a simple pole at AO means that M (A)

a

°-1 (A

(2.8)

in a neighborhood of AO' that ok

~

From the symmetry relation of Theorem l(iv), we know

0; the matrices ok are size n

point of the spectrum o(T).

x

n.

From part (i), AO is an isolated

This shows that AO E peT).

On the other hand, if

Ao E peT) then part (i) ensures that AO is an isolated singularity of Ma'

From

Theorem l(iii) we know that the diagonal entries of Ma belong to the PickNevanlinna class; i.e., {1m (A)} • {1m (Ma)kk(A)} > O.

Thus the diagonal entries

can have simple poles at most at any isolated singularity. M (A) - M (i) a

a

=

(Ie - i)

Now the identity

Jb1'~'~(t,I) A(t) I¥(t,i)dt a

may be used to bound the off-diagonal entries by expressions involving the (Ma)kk'

Indeed, the Cauchy-Schwarz inequality gives

If we multiply this by v, where A = AO + iv, and note that iv(M ) . . (A Ci.

remains bounded as v

+

]]

O

+ iv) i

0, then we may conclude the same about (M )'k' i.e., all Ci.

]

entries have simple poles at most. To identify eigenfunctions at points AO E peT) we again use (2.6). this time yields

Letting v

+

0

228

DON B. HINTON and K. SHA TV

from which it follows that the columns of $(t,AO)o_l belong to L!. columns are eigenfunctions.

Thus the

The number of linearly independent eigenfunctions

clearly equals the rank of the residue 0_1' If A = AO E peT), the nonhomogeneous problem (2.9) 2

->-

where g E LA' obviously does not have unique solutions, due to the presence of eigenfunctions. if

g is

Nevertheless, it can be proved that (2.9) can be solved uniquely

orthogonal to the manifold generated by {$(o,AO)o_l}'

In terms of oper-

ator theory, the operator T admits a "reduced" resolvent defined on the orthogonal complement ErL! 8{-

E Er LA 9{, (2.10) becomes [R\(T)f](x) ->-

[8(X,A) + ¢(x,\)M (\)] JX $1«t,i) A(t) f(t)dt a a

+ $(x,\) I

b* X

-

¢(X,AO)o_l

[8(t,i) + $(t,i)M (i)l'~A(t) f(t) dt a

Putting Ma(\)

MaCA) -

-1

l (\ - \0) be expanded further to [R\ (T)f] (x) = =

CJ_

-

and -

a

\0 in the third line above formally gives the expression (where a

subscript \ denotes partial differentiation)

229

Arguing as in [2] we can show that this integral actually converges and that the limit as A + AO may be taken as indicated.

Similarly, multiplying and dividing

by A - AO in the last line of (2.11) and using standard Legesgue theory arguments gives for the last integral -~A (x,A ) f~* [;t"(t,AO)O_l]* A(t) fet). O The limit of the second line of (2.11) is

-7-

Thus letting A + AO in (2.11) and using the orthogonality condition on f leads to the expression -+

-+

X

-+.1..

7-

[8(x,A ) + (x,AO)oO] fa V(t,A ) A(t) f(t) dt O O

+ ;t"(x,AO)o_l

fX

a

¢~(t,AO) A(t) f(t) dt

(2.12 )

which is formally the correct expression for the reduced resolvent.

That (2.12)

may rigorously be identified with the resolvent operator follows from standard operator theory arguments.

We omit the details.

If liw vM (AO + iv) = S ~ 0 and M (A) v+u a a is not analytic at AO' then we cannot have AO E peT) as otherwise

We briefly discuss (iv) and (iii). -1

is(\ - Ao)

lim vM (AO + iv) = O. We cannot have AO E peT) for in that case we would have v+O a -1 S = -io_ and analyticity of Ma(A) - 0_1 (A - AO) Thus AO E E(T) and we have l 2 only to demonstrate existence of eigenvalues. That (. ,AO)SE LA serves this purpose follows "from a modification of the analogous part in [2]. The proof that AO

~

Thus AO E PC(T).

PC(T) implies the conclusion of part (iv) also follows as in

[2], and we omit these details. As for part (iii), let AO E C(T). exhaustive subsets of E(T).

By definition C(T) and PC(T) are exclusive and

Hence Ma(A) cannot be analytic at AO.

In fact,

Ma (A) - is(A - AO)-l cannot be analytic at AO for any S as otherwise the singularity would be isolated.

Therefore the condition VMa(A

O

+ iv)

+

S

~

0 is ruled

out by (iv). If M (A) is not analytic at AO' then AO ¢ peT). If lim vM (AO + iv) = 0 then a v+O a Ma(A) cannot have a pole at AO since the value of the limit is the residue times

230

-i.

DON B. HINTON and K. SHA W

Thus AO E PC(T) U C(T).

The limit condition vMa(A

O

+ iv)

0 excludes PC(T)

+

by part (iv). We close with a few remarks on invariance of the spectrum.

First, it is possible

to compare Ma(A) functions arising from different choices of the matrices aI' a of (1.5).

2

It is simplest to do this through the special choice a

and the corresponding function which we denote simplY by M(A).

= In' a = 0 l 2 Let us write

o/a(X,A) and o/(x,A) for the corresponding unique L! solutions constructed in Theorem 1.

Invoking the limit-point hypothesis, the number of independent L!

solutions requires that 'r(X,A)

0/a (x,A)

C , where C is an n

x

n nonsingular

*

*

For x = a we obtain I [a~ - a; Ma (\)]C and M(A) = [a + a l Ma CA) ]C. 2 The first of these equations implies that [a * - a * Ma(A)] is invertible for all 2 l A, and the second therefore gives

matrix.

M(\)

=

[a* + a* M (A)][a* - a* M (\)]-1. 2

1

Ci.

1

2

a

This is analogous to a linking formula of Chaudhuri and Everitt [2]. Following the argument of [2], we may now establish the following invariance principles for spectra of operators T

Ci.

choices of matrices aI' at x

=

Ci.

and T

y

arising from different admissable

and Y , Y which determine the boundary condition 2 l 2

a: p (T )

a

U P (T )

a

p (T )

Y

U P (T ); Y

REFERENCES [1]

F. V. Atkinson, "Discrete and Continuous Boundary Problems," Academic Press, New York, 1964.

[2]

J. Chaudhuri and W. N. Everitt, On the spectrum of ordinary second order

differential operators, Proc. Royal Society Edinburgh 68A (1967-68), 95-119. [3]

W. N. Everitt, Fourth order singular differential equations, Math. Ann., 149 (1963), 320-340.

[4)

W. N. Everitt, Singular differential equations, I; the even order case, Math. Ann., 156 (1964), 9-24.

[5]

W. N. Everitt, Integrable-square, analytic solutions of odd-order, formally symmetric, ordinary differential equations, Proc. London Math. Soc. (3), 25 (1972), 156-182.

(6)

W. N. Everitt and C. Bennewitz, Some remarks on the Titchmarsh-Weyl m-coefficient, in: Tribute to Ake Pleijel, Department of Mathematics, University of Uppsala, Sweden, 1980.

TITCIIM-IRSlllt'L:YL THEOR Y

231

[7]

W. N. Everitt and K. Kumar, On the Titchmarsh-Weyl theory of ordinary symmetric differential expressions I: The general theory, Nieuw Archief Voor Wiskunde (3), 24 (1976), 1-48.

[8]

W. N. Everitt and K. Kumar, On the Titchmarsh-Weyl theory of ordinary symmetric differential expressions II: The odd-order case, Nieuw Archief Voor Wiskunde (3), 24 (1976), 109-145.

[9]

D. B. Hinton and J. K. Shaw, On Titchmarsh-Weyl M(A)-functions for linear Hamiltonian systems, J. Diff. Eqs., to appear.

[10]

D. B. Hinton and J. K. Shaw, On the spectrum of a singular Hamiltonian system, submitted.

[11]

V. I. Kogan and F. S. Rofe-Beketov, On square-integrable solutions of symmetric systems of differential equations of arbitrary order, Proc. Royal Soc. Edin. 74A (1974), 5-39.

[12]

B. M. Levitan and 1. S. Sargsjan, "Introduction to spectral theory: selfadjoint ordinary differential operators," English translation in Translation of Mathematical Monographs 39 (1975) (Amer. Math. Soc., Rhode Island, 1975).

[13]

R. M. Kauffman, T. T. Read, and A. Zettl, "The Deficiency Index Problem for Powers of Ordinary Differential Expressions," Springer-Verlag Lecture Notes in Mathematics vol. 621, Berlin, 1977.

[14]

P. W. Walker, A vector-matrix formulation for formally symmetric ordinary differential equations with applications to solutions of integrable square, J. London Math. Soc. (2),9 (1974), 151-159.

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Spectral Theory of Differential Operators I.W. Knowles and R. T. Lewis (eds.) © North·Holland Publishing Company, 1981

TWO PARAMETRIC EIGENVALUE PROBLEMS OF DIFFERENTIAL EQUATIONS C. Hunter Department of Mathematics and Computer Science Florida State University Tallahassee, Florida U.S.A.

Mathieu's equation and the angular spheroidal wave equation both lead to problems in which the eigenvalues a depend on a parameter q. The eigenvalues are analytic functions of q with simple branch points in the complex q-plane. Analytical, though approximate, relations between a and q are derived using asymptotic methods of WKBJ type. These relations appear to be valid uniformly throughout the complex q-p1ane. They predict the locations of the branch points, and reproduce a known result concer~ing the instability intervals of Mathieu's equation. MATHIEU'S EQUATION Mathieu's equation is d2y/de 2 + (a - 2q cos 2e)y = 0, (1) and its eigenfunctions are the solutions that are periodic of period 2n. For varying q, the operator of Mathieu's equation belongs to a selfadjoint holomorphic family of type (A) as defined by Kato (1966). When q = 0, the eigenvalues and eigenfunctions are simply a = n2 , y = cos or sin ne, n = nonnegative integer. For q f 0 , they form four separate classes because the cos 2e term in Mathieu's equation causes the Fourier series of the eigenfunctions to be composed of either cosine terms only or of sine terms only, and with arguments that are either even multiples of e only or odd multiples of e only. The four classes of eigenfunctions are therefore designated even cosine, odd cosine, even sine, and odd sine. The even cosine eigenfunctions, for instance, have the form

I

y(e,q) = A(2n)(q) cos k=O 2k

2ke, for a = a 2n (q) with a (0) = 4n 2 . 2n

(2 )

We shall follow the custom of using the symbols ak(q) and bk(q) for the eigenvalues of cosine· and sine eigenfunctions respectively, and identify eigenvalues by the square roots of their values at q = O. The eigenvalues are distinct when q is real, except when q = 0, and their behavior has been well studied (e.g. Meixner and Schafke 1954). SPHEROIDAL WAVE EQUATION The angular prolate spheroidal wave equation is 2 (d/drd [(1-n 2) dS/d n] + [" - c n2 - m2/(1_n 2 )] S

233

0, -1 " n ,,1.

(3)

234

CHRISTOPHER HUNTER

Here A is the eigenvalue and c 2 and m are two parameters. However, m arises as an angular wave number in applications, and is usually required to be a fixed non-negative integer. The boundary conditions on S are that it be finite at n = + 1. The transformation 2

_k

S (s i n e) 2 y, n = cos 8, C = 4q, A a + 2q - '" converts the spheroidal wave equation to d2Y/de 2 + [a - 2q cos 28 - (m 2 - '4) / sin 2eJ y = 0, 0,; 8 ] .

(14)

When the boundary condition y'(1;-;rr) = 0 is applied to the solution (13), one obtains the corrected, though still approximate, eigenvalue relation of sin [2h G( B)J = l:i exp [-2h H( B)J . (15a) A similar analysis of even sine eigenfunctions yields the eigenvalue relation sin [2h G( B )J = -1;-; exp [-2h H( B)J . (15b)

237

TWO PARAMlcTRIC EIGliNVALUE PROBLEMS

~~

~~.

TT-

: I

e

0

I

Fi gure 3 Stokes lines (full) and anti-Stokes lines (dashed) in the e plane for the case B

=

0.4.

0.8~----~--~----------r----------.--------~

13

I3 cn t I------\--+----\--+----~--I----~--+-I 0.4

o

2

4

6

h

8

Figure 4 Real eigenvalues for pure imaginary q plotted in the h - B plane. The branch points, at which iqi is maximum, are ringed. The full curves are exact, while the dashed curves are plotted from the asymptotic formulae (15).

238

CHRISTOPHER HUNTER

As Figure 4 shows, these approximate relations are accurate and allow adjacent eigenvalues to become equal. The branch points plotted in Figure 1 are points 2 at which Iql = B h is the greatest, and are ringed in Figure 4. The ringed points tend, with increasing h , to lie closer to the critical value S = B 't= cn 0.S811 at which H( B) = 0 but H' (S cn't) < O. The significance of this critical value of B is as follows. The turning point 8 0 lies far above the real o-axis when S is small, but descends towards this axis as S increases. When B = Bcrit ' the two downgoi ng anti -Stokes 1i nes from 8 0 pass through 8=0 and 8 = 1;;n. Now H( B) > 0 when B < Bcrit and is large when B is small. When the exp [-2h H( B )J terms in relations (lS) are insignificant, cosine and sine eigenvalues are indistinguishable in Figure 4. But exp [-2h H( B)J is 1 when S = Bcrit' while real solutions of (lSa) and (lSb) cease to be possible when it exceeds 2. Thi s happens for B only s 1i ght ly in excess of Bcri t when his large. ASYMPTOTIC ANALYSIS OF MATHIEU'S EQUATION:

THE GENERAL CASE

The analysis that is needed for general values of q in the upper half q-plane is also based on the use of WKBJ solutions and the turning point 8 , defined 0 now as the root of a = 2q cos 28 0 for which 0 < Re(8 0 ) < 1;;rr, again plays a crucial role. Although this turning point is no longer constrained to lie on the line Re(e) = 'on, the configuration of Stokes and anti-Stokes lines from it may still be such that a single Stokes line from eO intersects the real 8-axis between e = 0 and 0 = 1;;rr. Assuming this to be the case, and applying Heading's rules at the Stokes line and the relevant boundary conditions at 8 = 0 and 8 = 1;;rr , the following eigenvalue relations are obtained for the four classes of eigenfunctions of Mathieu's equation: even cosine, (16a) sin 1;; exp [-JJ (16b) exp [-JJ sin even sine, -~ -~; exp [-JJ cos odd cosine, (16c) exp [-JJ (16d) cos odd sine, ~i The quantiti es and J , which replace the 2hG and 2hH terms in (lS), are defined by the following integrals:

J

~'IT

o (a-2q

cos 2¢)

~ 2

d¢ = I(a,q),

rOo " J (a-2q cos 2¢) 2 d¢ = 1;;[l(a,q) + iJ(a,q)J.

o

(17)

(The location of the turning point and its Stokes lines depend on the eigenvalue, so that it is necessary to check our earlier assumption for self-consistency.) The general eigenvalue relations (16a-d) are effective in locating the other branch points. Further, these other branch points, like those on the imaginary q-axis, lie close to critical stages, where critical stages are defined now as occurring when: (i) (ii)

Anti-Stokes lines from the turning point

0 go to both 8 = 0 and e = 1;;n. The appropriate one of the eigenvalue relations (16) is satisfied. 8

These two requirements restrict the possible values of and J; instance, must be an integer multiple of "i for even eigenvalues.

J, for The critical

239

1'1\'0 PARAMETRIC EIGENI'ALUE PRORLl'AIS

stages must ultimately be located numerically because expressions for and J as functions of a and q involve complete elliptic integrals. However, some elementary analytical approximations for I and J , which technically are valid for small values of (q/a) only, are effective both in providing good initial approximations for the iterative determination of critical stages and, more importantly, in providing a true accounting of the pattern of branch points as shown in Figure 1. [See equations (22) below.J The branch points must also be located numerically. The critical stages provide excellent initial approximations for their iterative determination, and are indistinguishable from the branch points in Figure 1. Relations (16) remain useful as iqi ~ = , because the assumption about the turning point and the Stokes line that was made in deriving them remains valid. In fact, as iqi ~ = , 8 0 tends either to 0 and 1;'11. To see what is happening analytically, consider the even cosine relation (16a) for instance, and rewrite it in the form: (18) exp [iIJ - exp [-iIJ = i exp [-JJ The right hand side is small when (q/a) is small. All three terms of (18) are significant at the intermediate values of q at which branch points occur. As iqi ~ = however, one or other left hand side term becomes negligible. If the second of these is negligible, then exp [i(I-iJ)1 - i , 8 0 ~ ),11, and a - -2q as in a standard large-q asymptotic expansicn (Meixner and Sch~fke 1954). Similarly, when the first term of (18) becomes negligible, then exp [-i(I + iJ)J ~ -i , 8 ~ 0 , and a ~ 2q. Relations (16) therefore appear to be valid uniformly 0 throughout the upper half q-plane. This is possible because iq/ai is never large when a is large, and our basic assumption is that a is large. ESTIMATE OF THE INSTABILITY INTERVAL Although exp [-JJ is technically asymptotically negligible when (q/a) is small, yet retaining it allows the small difference between the cosine and sine eigenvalues of the same order to be estimated correctly. This difference corresponds to an instability interval when q and a are real. It can be computed, for either even or odd eigenvalues, on the basis of relations (16) using the approximati ons I = !:i1l ak2 [1 + O(q 2/a 2 )J , J = a '.2 Iw(4ia/q) -2J[1 + O(q 2/a 2 )J . (19) The difference in the exp [-JJ terms in the equations for the cosine and sine eigenvalues an and b both of which are almost n2 , gives n an - b - 4nin n 11

exp [-JJ

~ 4n [~Jn 4n2

(20)

11

This result is the large - n form of an exact result due to Levy and Keller (1963). It has also recently been obtained by Harrell (1981) in an asymptotic study of instability intervals. ASYMPTOTIC ANALYSIS:

THE SPHEROIDAL WAVE EQUATION

The asymptotic analysis of the spheroidal wave equation is also based on the approximation that a is large, and that iq/ai and m2 do not exceed 0(1). Hence, to a sufficient degree of approximation, the spheroidal wave equation reduces to Mathieu's equation except near the ends 0 and 11 of the range of e.

240

CHRISTOPHER HUNTER

Another approximation must be used in the regions near these ends. When this is done, the eigenvalue relations (16b) and (16d) generalize to sin [I - ~ mTI + \n] -~ exp [-J + iTI(\ - ~ m)] (2la) and cos [I - ~ mTI + \n] = y,i exp [-J + in(\ - Y, m)J , (2lb) respectively. Critical stages may again be located from these approximate eigenvalue relations, and the exact branch points of the eigenvalues a(q) that lie near them can be computed numerically. The elementary analytical approximations for I and J that were mentioned earlier can be used to provide rough approximations to the critical stages. When used with the eigenvalue relation (21b) for instance, they predict critical stages at \q\

=

~ (4N + l~ + m)2, arg q

=

e

n(y, -

~N-+~mm+-l~/6)

(22a)

where N is any non-negative integer and M is any even integer in the range -2N ~ M ~ 2N , and also at \q\

=

42 e

(4N + 25 + m)2 , 6

arg q

=

Y, m+- 25/6 \ ) ' n ('~ - M 4N - +m

(22b)

where M is now any odd integer in the range -2N-l ~ M $ 2N + 1 . Both formulae correctly predict the outward movement and counterclockwise rotation of the branch points with increasing m that is seen in Figure 2. CONCLUDING REMARKS Further details of the work that is described here can be found in Hunter and Guerrieri (1981, 1982). This work is heuristic and formal proofs are lacking. Yet the directions for further studies, and ways in which presently available rigoroUS results can be improved, are clearly indicated. For instance, both Meixner and Schafke and Kato show that the radii of convergence of the small-q power series for the eigenvalues an and bn of Mathieu's equation, which are determined by the locations of the branch points, exceed (n-l) for n > 2. The asymptotic analysis and the numerical results shows the radii of convergence to 2 be O(n). This work has been supported in part by the National Science Foundation under grant MCS-7728148. REFERENCES [lJ

Blanch, G. and Clemm, D.S., The double points of Mathieu's differential equation, Math. Compo 23 (1969) 97-108.

[2]

Harrell, E. M., On the effect of the boundary conditions on the eigenvalues of ordinary differential equations, Amer. J. Math., (1981) in press.

[3J

Heading, J., Global phase-integral methods, Quart. J. Mech. Appl. Math. 30 (1977) 281-302.

[4]

Hunter, C. and Guerrieri, B., The eigenvalues of Mathieu's equation and their branch points, Studies in Appl. Math. (1981) in press.

TWO PARAMFJ'R1C EiGENVALUE PROBLEMS

241

[5J

Hunter, C. and Guerrieri, B., The eigenvalues of the angular spheroidal wave equation, Studies in Appl. Math. (1982) in press.

f6J

Kato, T., Perturbation Theory for Linear Operators (Springer, Berlin, 1966)

[7]

Levy, D. M. and Keller, J. B., Instability intervals of Hill's equation, Comm. Pure Appl. Math. 16 (1963) 469-476.

[8J

Meixner, J. and Schafke, F. W., Mathieusche Funktionen und Spharoidfunktionen (Springer, Berlin, 1954)

[9J

Stokes, G. G., On the discontinuity of arbitrary constants which appear in divergent developments, Trans. Cambridge Philos. Soc. 10 (1857) 106-128.

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Spectral Theorv of Differential Operators I.W Knowles and R. T. Lewis (eds.) © North-Holland Publishing Company. 1981

SCHRODINGER OPERATORS IN THE LOW ENERGY LIMIT: SOME RECENT RESULTS IN L2(R4) Arne Jensen Department of Mathematics University of Kentucky Lexington, Kentucky 40506 USA

For a Schrodinger operator H = -6 + V in L2(R4) results on H in the low energy limit are given in the form of asymptotic expansions of (H - s)-l as s ~ O.

Consider a Schrodinger operator H = Ho + V, Ho = -6, in L2(R4) with V(x) = O( lxi-B) as Ixl ~ 00, S > 2. We give results on the low energy behavior of H in the form of asymptotic expansions of its resolvent R(s) = (H - s)-l as s ~ O. The behavior of R(s) as s ~ 0 is strongly dimensionally dependent. Results in L2(Rm) are given in [lJ for m = 3 and in [2J for m ~ 5. The results for m = 4 are given here without proofs. All the necessary techniques can be found in [1,2]. Details of the lengthy computations are given in [3]. For some related results see also [4,5,6,7J. We use the weighted Sobolev space Hm,s(R 4 ) defined for any m, s E R by Hm'S(R 4 ) = {f E S' (R4) I II (1 - 6)m/2 (1 + x2)s/2 fIIL2 < oo}. Here S' (R4) denotes the tempered distributions. We write Hm,s instead of Hm,s(R 4 ); (', .) denotes both the inner product on L2(R4) and the natural duality ms -m - s m s m' s' ms between H' and H ' . B(H', H ' ) denotes the bounded operators from H ' m to H ' ,s' , with the operator norm. Consider first H o

=

-6.

R (d o

=

(H

0

- d- l is given by

sl/2 (1) 1/2 "IT 2"IT Ix - y I H 1 (s Ix - y I ) , where H(i)(z) is the first Hankel function. Here T: k(x,y) shows that the operator T has the integral kernel k(x,y). Using the expansion for the Hankel function one obtains formally

where the operators G~ are given as follows: J

::'43

244

ARNE JENSEN

GO: (41T 2 r l /x_y/-2; Gl " o. o 0 2 (41Tr (-4)1-j((j-l)l'jlr l , j .::.1, and let 'l'(j) denote the digamma

Define c. J function.

c.{'I'(j) + 1jI(j+1) + irr}/x_y/2 j -2 - 2c. In(~/x-y/)·/x_y/2j-2,

G~ J

J

J

_c./x_y/2 j -2.

Gl J

J

The precise result on R (t;) is: o LEMMA 1. We have in the operator norm on B(H- l ,s,H 1 ,-s') the expansion R

o as

1; ...,.

O.

~

~. t;J ( 1n

L L

(d

j"O k=O

Here ]l(0)

k

d kG. + 0 (I;

~

(1 n d]l

() 9,

J

J

0, ]l(9,) " 1 for 1 .::. 1, and s, s' must satisfy 1

0

if 9, "0: 0

2

if

S,

£ >

s'

>

1/2, s + s' > 5/2;

1: s, s'

>

2L

Assumption of V. V is multiplication by a real-valued function V(x) such.that V defines a compact operator from H1,0 to H- l ,8 for some B > 2. H " Ho + V is the iU~drati:l f~:m sum. compact operator from H' to H ' S

Note that for every s E R, V is a

An expansion for R(t;) (H - t;)-l can be obtained from Lemma 1 and R(d = (1 + Ro(dVrl Ro(d, if we can obtain an expansion for the operator (1 + Ro(t;)vt1. We have 1 + Ro(dV

If 1 + GOV is invertible in B(H l ,-s,H l ,-s), 0 < S < S - 2, 0 is said to be a o regular point for H. In this case 1 + Ro(t;)V is invertible for small t;, and the inverse can be found using the Neumann series. This leads to an expansion for (1 + R (t;)V)-l with explicitly given coefficients. o THEOREM 1. Let 0 be a regular point for H. Then 9, j. k ( R ( ln ~r)£) as I; ...,. 0, R(d" L L t;J (1 n d Bjk + 0,1; j=O k"O in B(H-l,s,Hl.- s ') for£ .::.1, S > 4£, and s, s' > 2£. The first few coefficients Let X " (1 + G~V)-l.

are explicitly given as follows.

B~ = XG~X*, B~ = -XG~VXG~X*, B~ B~ = XG~X* - XG~VXG~X*.

=

XG1x*-

XG~VXG~X*

-

B~

=

XG~, B~

XG~X*,

XG~VXG~X*,

If 1 + G~V is not invertible, some further results are needed before we can find an expansion for (1 + Ro(z;)V)-l. For 0 < s < 8 - 2 let

245

SCHROEDINGER OPIiR/1TORS IN THE LOW ENERGY LIMIT

M = {f E H1 , - s

-

I

(1 + GOV) f = O}. 0

M is independent of s in the given interval. Since G~V is compact, M = {OJ generically. Precisely, consider H(x) Ho + xV, x real. Then !i(x) = {OJ except for a discrete set of values of x. LEMMA 2. For 0 < s < 2 we ha ve M = {g E H1 , - s I (H o + V) gO}, LEMMA 3. Let u EM. Then u E L2(R4) if and only if 6. Then we have in B(H-l,s ,H l ,-s') R(z;) = - z; - 1 (a - 1n z;) - 1 < . ,~)~ + 0 (1 ) a is given by (y is Euler's constant) a = ni + 1 - 2y - (4nr2

If

Assume 6 as

1; ->-

>

THEOREM 3. Let 0 be an exceptional point of the second kind for H. and s, s' > 6. Then we have in B(H- l ,s,H l ,-s') R(r;) = _I;-lp + ln 1; P VG 1Vp + 0(1) 2 0 o 0 If dim(!i) ~ 2 and we can find ~ E ~ with

O.

an exceptoo compli-

THEOREM 4. Let 0 be an exceptional point of the third kind for H. Assume S and 5, 5' > 6. We then have in B(H- 1 ,5, H1 , - s' ) R( 1;) = - I; - 1Po - I; - 1 (a - 1n I; r 1 < . , ~)~ + lnl;poVG1Vpo + 0(1) -+

O.

In(~lx-yl)·(V~)(x)(V~)(y)dxdy.

o is said to be an exceptional point of the second kind, if dim(M) !i = PoL2, i.e., ~ consists of eigenvectors for eigenvalue O.

as z;

12

>

>

12

246

.1 RNE JENSEN

REMARKS. Expansions to any order with explicitly given coefficients can be found using the techniques from [1,2]. GenerallY,expansions to higher orders require larger Band s, s'. The above results can be used to derive results on the time-decay of the wave functions, and asymptotic expansions for the scattering matrix in the low energy 1imit. REFERENCES [1]

Jensen, A. and Kato, T., Spectral properties of Schrodinger operators and time-decay of the wave functions. Duke Math. J. 46 (1979) 583-611.

[2J

Jensen, A., Spectral properties o~ Schrodinger operators and time-decay of the wave functions. Results in L (R m), m ~ 5. Duke Math. J. 47(1980),57-80.

[3]

Jensen, A., Spectral properties o~ S~hrodinger operators and time-decay of the wave functions. Results in L (R). Preprint, University of Kentucky, 1980.

[4]

Murata, M., Scattering solutions decay at least logarithmically. Japan Acad. Ser. A Math. Sci. 54 (1978) 42-45.

[5]

Murata, M., Rate of decay of local energy and spectral properties of elliptic operators. Japan. J. Math. 6 (1980) 77-127.

[6]

Rauch, J., Local decay of scattering solutions to Schrodinger's equation. Commun. Math. Phys. 61 (1978) 149-168.

[7J

Vainberg, B. R., On exterior ell iptic problems polynomially depending on a spectral parameter, and the asymptotic behavior for large time of solutions of non- s ta tiona ry problems. Ma th. USSR Sborn i k 21 (1973) 221- 239.

Proc.

Spectral Theorv of Differential Operators I.W. Knowles and R. T. Lewis (eds.) © North·Holland Publishing Company. 1981

LONG-TIME BEHAVIOR OF A NUCLEAR REACTOR* Hans G. Kaper Applied Mathematics Division Argonne National Laboratory Argonne, IL 60439

A fundamental problem of reactor physics is the determination of the long-time behavior of the neutron population in a nuclear reactor. In particular, one is interested in the question whether the total neutron density has a purely exponential behavior as t ? "". We formulate this problem as an abstract Cauchy problem, show that the solution is given by a semigroup, and investigate the asymptotic behavior of the semigroup. 1.

INTRODUCTION

A fundamental problem of reactor physics is the determination of the asymptotic behavior of a nuclear reactor for large times. Inside a reactor (a hi gh 1y heterogeneous compos ite structure of many different materi a1s) neutrons The neutrons move about freely (i .e., are generated by fission processes. rectilinearly and with constant velocity) until they interact with a nucleus of the reactor material; in the course of an interaction a neutron may disappear entirely (absorption), it may change its velocity (scattering), or it may trigger a fission process, as a result of which one or more new neutrons appear. The relevant space and time scales are such that interactions can be viewed as localized and instantaneous events. The equation that describes the rate of change of the neutron density inside the reactor is a linear transport equation; the dependent variable is the neutron velocity distribution function (f). If n denotes the reactor domai n (a bounded open convex subset of It 3), and Sis the neutron velocity range (a ball or spherical shell centered at the origin in ~3), then f(x,~,t)dxd~ represents the (expected) number of neutrons in a volume element dx centered at a point x ( n whose velocities lie in a velocity element d~ centered at the velocity t; E S at time t. The linear transport equation is a balance equation for f over the element dxdt; about (x,~), (1.1 )

if = -

~x • t;f(x,t;,t) - h(x,t;)f(x,t;,t)

+

J k(x,t;+t;')f(x,t;',t)dt;', S

x

E

(1,

s

E

S,

t

>0

The first term on the right is the (spatial) divergence of the neutron flux, which represents the effect of the free streaming; the second term represents the loss due to interactions at x, h(x,~)dt; being the collision frequency for neutrons with the velocity in the range dt; about t; at the point x; the third term represents the gain due to interactions at x, k(x,~+~' )dt; being the (expected) number of neutrons emerging with a velocity in the range d~ about t; after an

*Joint work with C. G. Lekkerkerker (U. of Amsterdam, Neth.) and J. Hejtmanek (U. of Vienna, Austria). This work was supported by the Applied Mathematical Sciences Research Program (KC-04-02) of the Office of Energy Research of the U.S. Department of Energy under Contract W-31-109-Eng-38. 247

HANS C, KAFER

248

interaction of a neutron with the velocity ~' with a nucleus of the reactor material at x. With Eq. 1.1 are prescribed an initial condition,

(1.2 )

(x, 1;)

E

rlxS ,

and a boundary condition on arl. The boundary condition expresses the fact that no neutrons enter the reactor from outside ("zero incoming flux"); it may be formulated as (1.3) where Sx

f(x,l;,t)

0

= {I; E S: x

t

+ t[,

E

Q for some t

> oj,

X E

>

°

an.

The quantity of interest is the total neutron density inside the reactor, i.e., the integral ~ f(x,l;,t)dxdl;; in particular, its asymptotic behavior as t + 00. For practical purposes one wants to know under which conditions on the functions hand k the integral behaves like a pure exponential as t + "'. We might add that, for many reactor materials, the functions hand k vary rapidly with the neutron velocity: they may display resonances, etcetera. As we shall see, a satisfactory solution to this problem has not yet been given. Partial answers are available, 'and new results from the theory of strongly continuous semi groups of positive operators in Banach lattices are being applied.

$

In the next section we give the functional formulation of the reactor problem as an abstract Cauchy problem. In Section 3 we show that this abstract Cauchy problem is solved by a strongly continuous semigroup of positive operators. In the final Section 4 we discuss some results about the asymptotic behavior of the semigroup. Details of the proofs, as well as related results, can be found in our forthcoming monograph [1, Chapter 12J. 2.

FUNCTIONAL FORMULATION

Let Q be a bounded, open, convex subset of 11 3 , and let S be a ball of finite radius centered at the origin in 11 3 In this section we shall show that the initial-boundary value problem 1.1-3 leads to an abstract initial value problem for the function f: [0,00) + Ll(rlxS). (The choice of an Ll-space is a natural one in the present context, as f is nonnegative and its L1-norm gives the total number of neutrons inside the reactor.) We begin with the definition of the collisionless transport operator (-T), which corresponds to the first term in the right member of Eq. 1.1. Two technical difficulties arise: one because the expression (a/ax)"l;f is singular at 1;=0, the other because the boundary condition 1.3 involves only part of the range of the variable 1;. Let C O(QxS) be the space of all functions f that satisfy the conditions (i) supp fCQxSaB for some S2, a> 0, where Sas = {I;EIl3: a~ 11;1 ~ sl; and (i i) f admits a {B,E)-extension to QExS for some E > 0; here, QE is a E-neighborhood of Q, and a (B,c)-extension is a function fEE COO(QExS) whose restriction to rlxS coincides with f and which vanishes on each incoming ray up to a point inside Q (i.e., for each (x,l;) E QxS, let T = T(x,l;) denote the unique nonnegative number such that x-TI; E aQ; then there exists a 1'1 E (O,T) such that f,O<x-sl;,i;) = for all s > 11.) Let TO be defined in CB' O(QxS) by the express i on '

s

°

(2.1)

(x,l;)

E

QxS,

f

e

C~,o{QxS) .

LONG TIMF BEHA V[()UR OF A NUCLEAR REACTOR

249

s

Then \I+TO(AE[) is a bijective map of C O(QxS) onto itself. If Re\ ) 0, then (\I+T O)-l can be extended by continuity to a bounded linear operator RA in L1 (QxS), where (2.2)

RAg(X,~) =

T

e-ASg(x_s~,~)ds ,

f o

for almost all (x,~) E QxS. This operator R\ is injective; its inverse is the closure of AI+TO' so if we define T by (2.3)

T = R\

1

AI ,

-

then T is uniquely defined and T is the closure of TO' The second and third term in the right member of Eq. 1.1 give rise to bounded linear operators in Ll(QxS), provided h E L""(QxS) and hp E L""(QxS), where hp(x,~I) = 1 k(x,~ -A*, then a(-(T+A 1 )+A Z) contains finitely many points Ak (k=O, ... ,m) l!!. each right half-plane ReA> -A* + s (s > 0), each .2..!. these points ~ ~ eigenvalue of -(T+A1)+A Z with finite algebraic multiplicity, and n-tuples

of

positive

n [i~1 (WI(ti)A Z ): (t1,t Z,···,t n ) c

(4.3)

m

W(t)

I

1121

numbers

~

\ t tOk

e

e

Pk + Zn (t) (I -P) ,

k=O where IIZn(t)II = o(exp(-A*+s)t) ~ t -> 00; here Pk and Ok ~ the projection and nilpotent operator associated with Ak' ~ P = PO+" .+P k . The representation 4.3 can be sharpened if one can show that the semi group W is irreducible. In the present context, W is irreducible if there exists a to > 0 such that W(t) is positivity improving for each t ~ to' Indeed, if W is irreducible, then AO is a simple eigenvalue, the projection Po is positivity improving, and there exists as> 0 such that the real part of any other point of a(-(T+A1)+A Z) is less than AO-s. Thus, Aot (4.4) W(t) = e Po + Z(t)(I-P ) , O where Z = [Z(t): t ~ OJ is a semigroup in (I-P O)L 1 (>2 x S). Although the spectral bound of the generator of Z is strictly less than AO' one can only conclude that the type of the semi group Z is less than or equal to AO' as Z does not necessarily consist of positive operators. REFERENCES [lJ Kaper, H. G., Lekkerkerker, C. G., and Hejtmanek, J., Spectral Methods in Linear Transport Theory (Birkhauser Verlag, Basel, to appear) [ZJ Kato, T., Perturbation Theory for Linear Operators York, 1966) [3J Derndinger, R., (1980), Z81-Z93.

Ueber

das

Spektrum

(Springer Verlag,

positiver Generatoren,

Math.

Z.

New 172

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This Page Intentionally Left Blank

Spectral Theory of Differential Operators I.W. Knowles and R. T. Lewis (eds.) © North·Holland Publishing Company. 1981

REMARKS ON THE SELFADJOINTNESS AND RELATED PROBLEMS FOR DIFFERENTIAL OPERATORS Tosio Kato* Department of Mathematics University of California, Berkeley

This will be a partial survey, with some new results included, of recent results on (essential) selfadjointness problems and their generalizations for linear differential operators. The main topics will be: the (essential) selfadjointness of second-order elliptic operators with oscillating potentials; the (essential) selfadjointness of higher-order elliptic operators; characterization of the domain for nonnegative potentials; the m-accretivity and mdispersiveness of degenerate-elliptic operators of second order in LP (Rm). 1.

Introduction. This is a partial, rather incomplete survey of recent results on the problems of (essential) selfadjointness for linear differential operators and their generalizations (such as quasi-m-accretivity). It consists of a review of more or less random samples of those recent results which are known to me, together with various comments and remarks, including some new results of our own. I can only apologize for any possible omission of other important results. In section 2 I review the (essential) selfadjointness in L2(Rm) of secondorder elliptic operators with variable coefficients. The emphasis is on the global oscillatory behavior of the potential, rather than its local singularities. Section 3 discusses a recent definitive result, due to Leinfelder and Simader, on Schrodinger operators with singular vector potentials. In section 4, I consider second-order, degenerate elliptic operators with real coefficients, which need not be formally selfadjoint. The main problem is the essential quasi-maccretivity of the minimal operator and the quasi-m-accretivity of the maximal operator, in the real Banach space LP(R m), 1 < p < =. In section 5, I discuss the domain of the selfadjoint operators considered in section 2. The main question is whether or not the domain is the intersection of the domains of the "kinetic energy" part and the "potential energy" part of the operator. In the last section, I shall introduce some results on the essential selfadjointness of general higher-order, strongly elliptic operators on Rm, including the domain problem just mentioned. In Appendices I give a proof of a theorem stated in section 2, together with other technical remarks. 253

254

2.

TOSIO KATO

Second-order ell iptic operators. 2.1. Second-order operators of the form m

T

(2.1)

I

j,k=l

o.a·k(x)Ok + q(x), J J

0j=dj-ibj(x), dj=d/dx j , j=l,···,m, have been studied extensively. We want to discuss some of the recent results.

In this section we make these standing assumptions: a = a kj real-valued, (2.2) a EO Lip(R m), jk jk where Lip denotes the set of locally Lipschitzian functions. (2.3)

(2.4)

The matrix (a.k(x)) is positive-definite J m b· EO Lip(R ), real-valued.

J 1 m m (2.5) q = q+ - q_, 0 ~ q+ EO Lloc(R ), o ~ q_ EO Lloc(R ). We thus assume that q is locally semi bounded, to avoid technical complications. 00

Under these assumptions, T¢ makes sense for all ¢ EO COO = Coo(Rm) but need not o 0 be in L2 = L2 (R m). If q EO L~oc' then H EO L2 and we define Tmin to be the operator in L2 given by T . ¢ = T¢ with domain O(T . ) = Coo. mln

mln

0

We define T as the restriction of T with O(T ) as the set of all u EO L2 2 max max with Tu EO L. Here Tu is taken in the distri bution sense. To make sense out of 1 the term qu as a distribution, we assume that u EO O(Tmax) implies qu EO Lloc ' T in this general setting was considered in Kato [15J in the special case max 2 o'k' b. = 0 and with q = O(lxl ) (with mild local singularities). It was a jk J J shown that Tmax is selfadjoint in L2. Since (2.6) Tmax = T~in if q EO L~oc' 2 this proves also that Tmin is essentially selfadjoint if q+ EO Lloc ' These results have been generalized by many authors, including Eastham-EvansMcLeod [7], Frehse [llJ, Oevinatz [3J, Evans [8J, Knowles [20,21J, Kalf [14J, with more and more emphasis on oscillatory potentials q. All these papers contain a local characterization of O(T ), which says that -max 1/2 2 ) implies d.u, O.U, q+ u EO Ll ' (2.7) u EO O(T max J J oc and this is essential in the proof of the selfadjointness of Tmax or the essential selfadjointness of Tmin under various additional assumptions on the global behavior of q. A particularly strong and useful result is given by Knowles [22]. There is a densely-defined operator To in L2 such that (2.8) T C T* = T . o 0 max To is called the minimal operator by Knowles, but we shall reserve Tmin in the original sense. What is important is that Tmax is the adjoint of a certain symmetric operator. This is very effective in applications, since it reduces the

SELFADJOINTIVESS PROIH.t:JIlS H)R DIFFFRLN1T1L OPI]RATORS

255

proof of the selfadjointness of Tmax to the proof that (2.9)

(Tmax

+ i)u+

= 0

implies

u+ = O.

2.2. There are some variations among these authors in the continuity assumptions on the a. k and b .. For example, Eastham-Evans-McLeod assume that 1+ J J a. E C a rather than Lip, and are followed by Evans, while Frehse assumes only k L~P. I presume that Cl +a was technically required in connection with the local singularities of q. I would conjecture that Lip is sufficient even in the presence of such local singularities, though this would require a careful study. 2.3. Among these works listed above, it seems to me that the most general sufficient condition for the essential selfadjointness of Tmin is contained in Evans [8] (although this paper has the main purpose of considering the powers Tk ). I would rather not reproduce his condition here, which is not very simple, even in the special case (2.5) I am assuming. It will only be noted that it consists of the restriction of the growth rate of the a jk , expressed in terms of an upper bound p+(r) of the largest eigenvalues of (ajk(x)) for Ixl = r, to be correlated with the growth rate of q+ in a complicated way. The condition is general enough to allow a variety of oscillatory behaviors of q. It should be noted that the assumptions of Evans imply (2.10) Joo p(rf 1/2 dr=oo, 1

+

(see Appendix 3)

although this is not explicitly mentioned in the paper. It is somewhat disturbing, in view of the otherwise very general nature of his condition. Looking into other papers, I found that very few authors gave sufficient conditions that do not imply (2.10). Frehse [11] is one of the few, and his condition regarding the growth rate of p+(r) is very mild. But he had to correlate it with P_ (r), a lower bound of the smallest eigenvalues of (ajk(x)) for Ixl = r, even when q_ = O. I am rather reluctant to introduce p_(r) into the assumptions when q has no local singularities. Actually Evans also uses p_{r), but only in connection with such local singularities of q. 2.4. Some comments are in order regarding (2.10). An analogous but stronger condition (in which p+(r) is replaced with a*(r), the supremum of the largest eigenvalues of ((a.k(x)) for Ixl ~ r) was implied by the assumptions used in J .. Ikebe-Kato [12], as was pointed out by Jorgens [13]. Jorgens was able to remove this defect, but not very substantially. In fact in Ikebe-Kato, one could have replaced a*(r) with the radial bound of (ajk{x)): m -2 a* (r) = sup arad{s), (2.11 ) I aJ·k{x)xJ.xkr sup rad l<s S. It is known that {e- } is a quasi-contractive semigroup on LP if and only if A is quasi-m-accretive. 00.

Thus one may raise the questions: Is T quasi-m-accretive in LP? min Is Tmax quasi-m-accretive in LP? It is a priori conceivable that the answer is yes or no for both questions, or yes for one question and no for the other. These questions are related to the p' same ones for the formal adjoint 5 of T. If we consider Smin and Smax in L (01) and (02) are equivalent (in the reversed order) to (01') Is Smin quasi-m-accritive in LP '?

(01) (02)

(02')

Is Smax quasi-m-accretive in LP'? Indeed, (01') is dual to (02) and (02') to (01) by the well-known relations (4. 5)

5

= T*.

T

= 5*. .

max mln' max mln It was shown by Devinatz [5J by probabilistic methods that the answers to these questions are yes if the coefficients a· k , a., and a have compact supports. I J J conjecture that the same is true in the general case of (4.2-3) so that one has T = T. for all p, but so far we have proved this only for (01) and (02') max mln with p ~ 2 and (02) and (01') with p ~ 2. (In any case all can be proved if oo one adds the condition da jk E L . ) 4.2. In addition to the quasi-accretivity of Tmin and related problems, there is another important notion attached to the operator T. One may ask whether or not -T is guasi-dispersive. According to Phillips [25J, a linear operator -A in LP is dispersive if (4.6) (Au,u~-l)~O for U E D(A), where u+ = max{u,O}. We shall say -A is quasi-dispersive if -A - S is dispersive for some constant s. Again, -A is quasi-m-dispersive if, in addition, the range of A + A is the whole space LP for A > s. (Actually Phillips defines dispersiveness in general

260

TOSIO KATO

Banach lattices.) According to a theorem of Phillips [25], a densely-defined dispersive operator -A with nonempty resolvent set is m-dissipative (i.e., A is m-accretive) and, in addition, the semi group e- tA is positivity-preserving. In the case of our operator (4.1), it is expected that -Tmax is not only quasi-m-dissipative but also quasi-m-dispersive, so that the semigroup generated is positivity-preserving. Since (4.6) is similar to the corresponding accretivity (dissipativity) condition (4.4), the same computation can be used to acquire this additional information. 5.

The domain characterization. Another problem related to (2.1) is an explicit characterization of the domain of T For example, consider the Schrodinger operator max (5.1) T = -t, + q(x). Given a q such that Tmax is selfadjoint, one may ask if O(T ) = D(-t,) n D(q) = H2(Rm) n O(q). (5.2) max Results of this kind are important in many problems. In the theory of evolution equations, for example, it is important to construct an isomorphism S of a Banach space Y, continuously embedded in another Banach space X, onto X. S =T is a good choice for Y = H2 n O(q) and X= L2 if (5.2) is true. max Questions of the form (5.2) have been studied by Sohr [30,31]. A convenient theorem due to Sohr is the following. Let A, B be m-accretive operators in a -1 Hilbert space H, with A bounded. Then A + B with O(A + B) = O(A) n O(B) is m-accret i ve if -1 2 for U E O(B*), Re(B*u,A u) > - c~u~ (5.3) where c


o. Theorem I is applicable with W= 1, V = U = log(l + Ixl) provided P - 2:5.- (l-Tj)o. Since n can be arbitrarily small, it suffices that p < 2 + o. Thus any fast growth rate p for a rad is admissible if q+ grows fast enough. I do not know whether or not p = 2 + 0 is allowed, though it is all right if m = 1. Appendix 3. We sketch a proof that the assumptions in Theorem 1 of Evans [SJ imply (2.10). First we note that they imply, among other things, p:/2w' ~ K(l + 0~/2w), (A12 ) (r .:: 1) (Al3)

J

p-l/2(1 + 0~/2w)w dr

l

(A13) is exactly condition (iv) in [SJ, and (A12) follows directly from (i) there. Now (2.10) is obviously true if condition (v-a) of [8J is assumed. If, instead. (v-b) is assumed, then 01 is bounded. In this case suppose (2.10) is not true. Then it follows easily from the differential inequality (A12) that w is bounded, hence that the integral in (A13) is finite--a contradiction.

CORRECTIONS AND SUPPLEMENTS 1. In section 3.1, it was incorrectly implied that the result h max h. mln under assumptions (3.2) was due to Leinfelder-Simader [23]. Actually the same result had been given by Simon [29]. 2. The conjecture in Section 4.1 has been proved. 3. It has been shown that in Section 6, condition (6.5) can be weakened to (6.6).

SELE4DjOININESS PROBLLMS l'OR DIFFERENTI.4L OPERATORS

265

REFERENCES [1]

Browder, F. E., Functional analysis and partial differential equations, II, Math Ann. 145 (1962), 81-226.

[2]

Cordes, H. 0., Self-adjointness of powers of elliptic operators on noncompact manifolds, Math. Ann. 195 (1972), 257-272.

[3]

Devinatz, A., Essential self-adjointness of Schrodinger-type operators, J. Functional Anal. 25 (1977), 58-69.

[4]

Devinatz, A., Selfadjointness of second order degenerate-elliptic operators, Indiana Univ. Math. J. 27 (1978), 255-266.

[5]

Devinatz, A., On an inequality of Tosio Kato for degenerate-elliptic operators, J. Functional Anal. 32 (1979), 312-335.

[6J

Dung, N. X., Selfadjointness for higher-order elliptic operators, Dissertation, University of California, Berkeley, 1981.

[7J

Eastham, M. S. P., Evans, W. D., and McLeod, J. B., The essential selfadjointness of Schrodinger-type operators, Arch. Rational Mech. Anal. 60 (1976), 185-204.

[8J

Evans, W. D., On the essential self-adjointness of powers of Schrodingertype operators, Proc. Roy. Soc. Edinburgh 79A (1977), 61-77.

[9J

Evans, W. D., and Zettl, A., Dirichlet and separation results for Schrodinger-type operators, Proc. Roy. Soc. Edinburgh 80A (1978), 151-162.

[10] Everitt, W. N., and Giertz, M., Inequal ities and separation for Schrodinger type operators in L2(Rn), Proc. Roy. Soc. Edinburgh 79A (1978), 257-265. [11] Frehse, J., Essential selfadjointness of singular elliptic operators, Boletim da Soc. Brasil. de Mat. 8 (1977), 87-107. [12] Ikebe, T., and Kato, T., Uniqueness of the self-adjoint extension of singular elliptic differential operators, Arch. Rational Mech. Anal. 9 (1962), 77-92. [13] Jorgens, K., Wesentliche Selbstadjungiertheit singularer elliptischer Differentialoperatoren zweiter Ordnung in Co(G), Math. Scand. 15 (1964), 5-17. [14] Kalf, H., Gauss's theorem and the self-adjointness of Schrodinger operators, Arkiv. for Mat. 18 (1980), 19-47. [15] Kato, T., A second look at the essential selfadjointness of the Schrodinger operators, D. Reidal Pub. Co., Dordrecht 1974, 193-201. [16] Kato, T., Remarks on Schrodinger operators with vector potentials, Integral Equations and Operator Theory 1 (1978),103-113. [17] Kato, T., On the Cauchy problem for the (generalized) Korteweg-de Vries equation, to appear. [18] Keller, R. G., The essential self-adjointness of differential operators, Proc. Roy. Soc. Edinburgh 82A (1979), 305-344. [19] Keller, R. G., The essential self-adjointness of differential operators with positive coefficients, ibid. 345-360.

266

TOSIOKATO

[20] Knowles, r., On essential self-adjointness for singular elliptic differential operators, Math. Ann. 227 (1977), 155-172. [21] Knowles, I., On essential self-adjointness for Schrodinger operators with wildly oscillating potentials, J. Math. Anal. Appl. 66 (1978), 574-585. [22J Knowles, I., On the existence of minimal operators for Schrodinger-type differential expressions, Math. Ann. 233 (1978), 221-227. [23] Leinfelder, H., and Simader, C. G., Schrodinger operators with singular magnetic vector potentials, to appear. [24J 01einik, O. A., Linear equations of second order with nonnegative characteristic form, Mat. Sb. 69 (111) (1966),111-140; AMS Translation Ser. 2, vol. 65 (1967), 167-199. [25]

Phillips, R. S., Semi-groups of positive contraction operators, Czechoslovak. Math. J. 12 (87) (1962),294-313.

[26]

Read, T. T., A limit-point criterion for expressions with intermittently positive coefficients, J. London Math. Soc. (2) 15 (1977), 271-276.

[27]

Schechter, M., Essential self-adjointness of the Schrodinger operator with magnetic vector potential, J. Functional Anal. 20 (1975), 93-104.

[28]

Simon, B., Schrodinger operators with singular magnetic vector potentials, Math. Z. 131 (1973),361-370.

[29]

Simon, B., Maximal and minimal Schrodinger forms, J. Operator Theory 1 (1979), 37-47.

[30]

Sohr, H., Uber die Selbstadjungiertheit von Z. 160 (1978), 255-261 .

[31]

Sohr, H., Uber die Existenz von Wellenoperatoren f~r zeitabh~ngige Storungen, Monatsh. Math. 86 (1978), 63-81.

Schr~dinger-Operatoren,

..

* This work was partially supported by NSF Grant MCS-79-02578.

Math.

Spectral Theory of Differential Operators I.W. Knowles and R. T. Lewis leds.} © North-Holland Publishing Company, 1981

A WEYL THEORY FOR A CLASS OF ELLIPTIC BOUNDARY VALUE PROBLEMS ON A HALF-SPACE Robert M. Kauffman Department of Mathematics Western Washington University Bellingham, Washington, U.S.A.

A Weyl theory for a class of elliptic partial differential operators on a half-space is developed, and related to the well-posedness of the boundary value problem Mf

=

g, where f and g are square-

integrable on the half-space and f is required to satisfy certain conditions at the boundary.

The

operators discussed are natural generalizations of ordinary differential operators with positive coefficients.

Recent developments in the Weyl

theory for these ordinary differential operators are reviewed, and the ODE results are related to the PDE results. O.

INTRODUCTION

Consider the following problem:

"Solve the equation Mf

=

g on the

open region U, with g in L (U), where M is an elliptic partial dif2 We deal for simplicity with the case

ferential operator on U."

R~ = {xix = (xl"" ,xk ) and Xl > a}. We assume that the coefficients of M are restrictions to R~ of elements of COO (Rk), and are positive in a certain sense. We ask what con-

where U is the half-space

ditions must be imposed on f to make the problem well-posed. If one examines physical situations, such as the heat equation or the Schroedinger equation, where this problem arises, it becomes clear that one should expect to impose conditions at the boundary Xl = 0 and should also in many applied situations require f(x) to become small as Ixl becomes large. A very reasonable smallness condition is the condition that f be in

L2(R~).

into a problem in the Hilbert space L2(R~). stated problem in

This turns the problem Once one solves the

L2(R~), one may often use sernigroup theory to solve

related parabolic or hyperbolic problems such as

d2~/dt2

=

-M~. 267

3~/dt =

-M~

or

R.M. KA UFMANN

268

Having decided to impose these conditions on f, we must worry about two things.

First, we must ask whether the solution is uniquely

determined.

Second, we must ask whether we have imposed so many con-

ditions that, for some g, no solution exists.

The second worry turns

out to be groundless; there is always a solution when M and the boundary conditions at xl =

°

satisfy certain reasonable hypotheses.

However, even in situations where M and the boundary conditions appear very innocent, the solution may fail to be unique.

This un-

expected non-uniqueness occurs, when it occurs, because additional boundary conditions at infinity upon f are necessary to specify the solution; merely requiring f to be in

L2(R~)

is not enough.

We examine the question of boundary conditions at infinity both in the ODE and PDE case.

We relate the question to the essential self-

adjointness of a certain operator in L (R:), and examine conditions 2 on the coefficients to guarantee that the operator is essentially self-adjoint, and hence that the solution is unique.

This essential

self-adjointness is important in its own right because it means that in a certain sense the problem on the infinite region may be approximated by using finite regions. 1.

THE ONE-DIMENSIONAL CASE

In this section we examine the one-dimensional case, which is a pro~ lem in ordinary differential equations.

We change the region from

R! = (0,"') to (1,"') to make the statement of some of the theorems easier.

We first state the problem precisely.

Problem P.

Given g in L (1,"'), find an f in L 2 (1,00) such that Mf=g, 2 and such that (f(l), f(l) (1), .... ,f(2N-l)(1» is in S, where S is an 2N N-dimensional subspace of complex 2N-space K ,and where M is a 2Nth order differential expression.

We assume that M and S satisfy

the following: i) M = L~(-l)jDjPjDj, with D = djdx, and with each Pj the restriction to (1,"') of an element of C"'(-"',"'); ii) Pj ~ 0 on (1,"'); iii) PO ~ E > 0 on (1,"'); iv) PN > 0 on [1,"') where we have defined PN(l) by using the continuous extension of PN to [1,"'); 2N v) if f and g are in C [1,"'), with f(x) = g(x) x large, and with (f(l), f(l)(l), ... ,f(2N-l)(1» in Sand (g(l), g(l)(l), ... ,g(2N-l)(1»

in S, we assume that

JiMfg = L~ JiPjf(j)g(j)

0 for

269

Remark: Mf

=

To write M

=

L~(-l)jDjPjDj is to mean that

L~(-l)j(pjf(j))(j).

Remark:

Assumption v) above is a requirement upon S.

fied in many cases of interest for applications. satisfied if S is the set of all vectors v v.~ = 0 for i 0, and Po arbitrary. Theorem 1.8.

(Kauffman [5], 1977).

Let M

=

i~(-l)jDjPjDj.

Suppose

each Pj is a finite sum of real multiples of real powers of x.

Then

the solution to problem P is unique when degree Pi - 2i hits its maximum value for only one i. exist examples when N

It is also unique when N

=

2.

There

3 of M where the solution to problem P is not unique; such examples may be found with M = _D 3 x a D3 + (:ix o - 6 , =

for certain values of a > 6 and (:i > O. Theorem 1.9.

(Kauffman [6], 1980).

Let M

=

Z~(-l)iDipiDi

Suppose

that the Pi satisfy certain regularity hypotheses, that PN ~ a > 0, and that, for some j, Pi = O(p~-a) for some a > 0 and all i j. J n Suppose in addition that for all n, x = O(p.). Then the solution

+

to problem P is unique. that p~j) l

=

J

One of the chief regularity hypotheses is

O(p~+1 + 1) for all positive I and j, although certain l

other technical hypotheses are necessary.

The regularity hypotheses

are satisfied, for example, if all Pi are finite sums of terms of the form fe g , where f and g are finite sums of real multiples of real powers of x. Remark:

The moral of theorems 1.8 and 1.9 seems to be that problem

P is well-posed if any coefficient may be regarded as the biggest, in the sense of these theorems, provided the coefficients are sufficiently regular. Remark:

Question 1.2 is still unresolved if N

=

2.

A related re-

sult of considerable interest is announced by T. T. Read in these Proceedings.

271

.1 It'/:YL THEORY

2.

THE PDE CASE

We now study the case of an elliptic partial differential operator k on the half space R+ {xixI > O}, where x = (xI,x2' ... ,xk ) is an element of Rk. Guided by the case of an ordinary differential operator, we I et M .

=

.

k

.

_.N

j

j

j

j

LIM i , wlth Mi - LOC-l) DiPijD i , where Di denotes N

...

.

.

dJ/dX~, and where by this notation, M.f = 2: (-1)J (lJ;aX~Cp .. C1Jf/dx~). 0 l l k l lJ l We assume that each p .. is the restriction to R+ of an element of k lJ C (R ), and that each Pij is non-negative, PiO ~ E > 0, and the con00

tinuous extension of each PiN to {xixI ~ O} is non-vanishing. Before we can phrase problem P, we need to deal with a new difficulty.

From the knowledge that f and Mf lie in L2CR~), one cannot

say anything about the behavior of f at the boundary of R~. (This is in sharp contrast to the ODE case.) Hence the imposition of boundary conditions is not possible unless we place more regularity requirements on f. Definition 2.1. We say that f is 2N-regular on R~ if, for all ¢ in Co CRk) , ¢f is in H2N CR~), where H2N CR~) is the Sobolev space of all functions g such that all partial derivatives of g of order up to and including 2N lie in L2(R~). in the distributional sense.) Remark:

(The partial derivatives are taken

The next lemma makes precise the notion of a boundary value.

Lemma 2.2. There exists a linear transformation T with domain the 2N-regular functions on R~ and with range contained in the set of ordered 2N-tuples of elements of L~ocCRk-l) such that, if Tf

=

ChO' ... ,h 2N - l ), the h. have the following properties: a) for any 8 in ~OCRk-I), 8h is in H2N-l-iCRk-I); i b) if f is in H2NCR~), then hi is in H2N-l-iCRk-l);

c) if fn is a sequence of 2N-regular functions on R~ such k that, for all ¢ in CO(R ), ¢f n converges to ¢f in H2NCR~), then, if Tf n (h nO ' h nl ,·· .hn2N - I ) and Tf = (h O ' ... ,h 2N - I ), it follows that, k-l for any 8 In CO(Rk-l ), 8 h . converges to Bh. in H2N-I-i CR); nl 2N k l d) if fn is a sequence in H CR+) and fn converges to f in H2N(R~), then Cusing the notation of part c) h ni converges to hi in •

00

H2N-I-iCRk-l) ; e) if f is the restriction to Rk of an element of C2N (Rk),

+

and Tf

=

(h O '.·· ,h 2N - I ), then

hiCx2'··· ,xk)

=

(li f / dx icO,x2'··· ,xk );

272

R.M. KAUFMANN

f) if Ch O"" ,h 2N _ l ) is an ordered 2N-tuple of complexvalued functions with each hi in C;CRk-l), then Ch "" ,h - ) = Tf O 2N l for some f which is the restriction to R~ of an element of CO(Rk ). Remark:

The content of the preceding lemma is that T is the continu-

ous extension, in a natural sense, of the restriction map defined in e).

The lemma is essentially well-known in the theory of PDE, al-

though a slight modification is needed to extend the usual trace mappings to the 2N-regular functions. Remark:

Now that we have defined what we mean by a boundary value,

we are ready to state problem P for the PDE case. Given g in L2(R~), find an f in L2(R~) such that f is k k 2N-regular on R+, Mf = g on R+, and Tf(x 2 , ... ,xk ) is in S for almost k l every (x "" ,x ) in R - , where S is an N-dimensional subspace of k 2 complex 2N-space K2N such that, for any f and g which are restric-

Problem P.

tions to

R~ of elements of C;CRk), with Tf(x 2 , ... ,xk) and

Tg(x , ... ,x ) in S for all points (x '" 2 k 2 N

j

.,X ) of Rk - l

k

j-

LO J k PljDlfDlg· R+ Notation 2.3.

Let W be the set of all f such that f is the restric-

tion to R! of an element of C;(Rk), with Tf(x 2 , ... ,x ) in S for all k k-l points (x2"" ,xk ) of R . Let R be the restriction of M to W. Let HR be the Friedrichs extension of R. Remark:

The following theorem is proved in Kauffman (to appear), al-

though it seems likely that a number of earlier writers, including Friedrichs, knew the result.

I felt it was necessary to give a

proof because I could not find an explicit reference. Theorem 2.4.

Every f in the domain of HR is 2N-regular, and

Tf(x , ... ,X ) is in S for almost every point (x 2 '" 2 k Hence, in particular, if f Furthermore, HRf = Mf.

k l k ) of R - . H-1 , f is a R

.,X =

solution to problem P. Remar~:

l~ HR =

R, where R is the operator theoretic closure of R,

then, for any f in the domain of HR , there is a sequence fn of elements of W such that fn converges to f and Mf n converges to HRf in L2(R~). This gives hope of computing things about HR by using compact support functions. Hence the question of when HR = R has some independent interest. Remark:

We now introduce two important properties.

273

A II'LTL l'HEORY

Property a.

The solution to problem P is unique.

Property b.

HR

Remark:

=

R.

(In other words, R is essentially self-adjoint.

We investigate the relationship between these two desirable

properties.

It follows from well-known theorems in ordinary dif-

ferential operator theory that they are equivalent in the ODE case. It is not hard to see that in the PDE case Property b implies Property a.

To go the other way, one first tries to study the or-

thogonal complement of range R.

Unfortunately, it is difficult to

find elements of this orthogonal complement which are regular enough to have boundary values.

Hence a more sophisticated argument seems

necessary. Theorem 2.5.

(Kauffman, to appear).

Let Rand H be as above.

Let Q

be the restriction of M to the set of 2N-regular f such that k-l Tf(x ,· .. x ) is in S for almost every point (x , ... ,X ) of R . 2 k 2 k These are equivalent: i) R is essentially self-adjoint; ii)

R=

iii) R

iv) HR

H , where R

R

is the operator-theoretic closure of R;

Q;

= =

Q;

v) Q is 1-1.

Furthermore, if R is not essentially self-adjoint, there exists an f such that Mf

=

0, f is in

L2(R~), all partial derivatives of f of all

orders are extendable to continuous functions on {x I xl :: O}, and (f, Dlf, ... ,DiN-If)

(0,x " , .xk ) is in S at all points (0,x 2, ... ,xk) 2 of the hyperplane xl = 0, where we have defined these partial derivatives at xl Remark:

=

0 by using their continuous extensions.

Property a is the same as Property v) of the theorem, and

Property b is the same as Propert i).

Hence Properties a and bare

equivalent. Proof of Theorem 2.5:

We give a brief sketch of the proof of theo-

rem.

R is contained in HR , it is clear that i) implies ii). Since integration by parts may be used to show that Q is contained in K",

Since

it is clear that ii) implies iii).

Since R is contained in H , and R HR is contained in Q, it follows that iii) implies iv). Since HR is 1-1, it is clear that iv) implies v). We now prove the only hard part of the theorem; we show that v) implies i).

Let F be the Friedrichs extension of R2.

It is possible,

274

R.M. KAUFMANN

with some effort, to prove that for any f in the domain of F, f is 4N-regular, and T(Mf)(x 2 , ... xk) is in S for almost every (x2"" ,xk) of Rk - l From the definition of the Friedrichs extension, it is clear that domain F is contained in domain

R.

If R is onto, it is self-adjoint, since it is symmetric.

If R is

not onto, then, since range R is closed, there is an element ¢ of COO (Rk) such that ¢ is not in the range of R. ¢ is clearly in vI. But,

o +

since F is onto, M¢ = Ff for some f. Q(¢ - Rf) to.

O.

=

But ¢ - Rf

f O.

Hence M¢ = Q¢ = Q(Rf).

Thus

Hence Q is not 1-1, if R is not on-

The proof of the equivalence of i) -v) is completed.

be shown that f is m-regular for all positive m.

It may

One may use this

fact together with Sobolev's imbedding theorem to prove the final assertion. Question.

What are conditions on the coefficients which guarantee

that R is essentially self-adjoint? Remark:

We answer the question for certain types of coefficients.

Our results apply to the whole-space case as well as the half space case, and are new for the half-space and higher-order whole-space cases.

In the whole-space case, we let R be the restriction of M to Our theorem

C~(Rk), and ask whether R is essentially self-adjoint.

contains no new assertions about the second-order whole-space case, as the specialization of our result to this case follows as a very special case of the strong second-order theorem announced by T. Kato in these Proceedings. Remark:

Our results are about coefficients which are like polynomi-

als, but are more general.

The virtue of this more general class is

that it permits arbitrary exponents and is translation-invariant. Definition 2.6. 2 f(x) c¢(x) (x

We say that f is in Z[a,oo) if

+ 1)A/2 + ~(x) + y(x), where c is a complex number

and i) ¢ and

~

are restrictions to [a,"') of elements of

ii) y is the restriction to [a,"') of an element of C~(-oo,oo);

iii) ¢(x) approaches 1 as x approaches infinity; (;) 2 -·/2 f or all j ::: 1; i v) ¢ J (x) = 0 (x + 1) J 2 v) ~(j)(x) = o(x + 1)(A-j)/2 for all j ~ 0; vi)

~

=

0 if c

=

O.

A is called the degree of f.

We take A

-00

if c

O.

275

A II'E1'1- l'HH)R Y

Definition 2.7. be in

z(-oo,~)

A complex-valued function f in Coo(_oo,oo) is said to

if

i) the restriction of f to [0,00) is in Z[O,oo); ii) if g(x)

f(-x), the restriction of g to [0,00) is in

=

Z[O,oo). Theorem 2.S. N

Mi

=

(Kauffman, to appear).

.,

.

~O(-I)JDfPijDf'

Let M

=

~~i'

where

Let R be as in Notation 2.3.

Assume the

following: i) for i > 1 and all j, Pij(x) = hijCx i ) for all x in k R+, where h .. is in Z(-oo,oo);

lJ

ii) Plj(x) = hlj(x l ), where h lj is in Z[O,oo); iii) if n(l,i,j) is the degree of the restriction of h ij to [0,00) for i ~ 1, and if n(2,i,j) is the degree of the restriction of the function gij(x i ) = hij(-x i ) to [0,00) for all i > 1, then n(l,i,j) 2j < n(l,i,O) for all i ~ 1 and all j > 0, and n(2,i,j)

2j < n(2,i,0) for all i > 1 and all j

°

> 0.

°

iv) Pij ~ for all i and j, PiO ~ E ~ for all i, and the continuous extension of PiN to Xl 2 is non-vanishing for all i.

°

Then R is essentially self-adjoint Remark:

It should be noted that Pij must be a "polynomial" in Xi

only, by hypotheses i) and ii). Remar~:

In the ODE case, any coefficient is allowed to be the big-

gest, where the size is measured by taking degree Pj - 2j.

In the

PDE case, we need PiO to be the biggest, using this measure of size. k

Theorem 2.8'.

Let M

N

..

.

LIM i , where Mi LO(-I)JDIPijDi. k Suppose each PiJ' is in Coo(R ). Let R be the restriction of M to k CO(R ). Assume the followin?: =

00

i) for all i and j, Pij (x)

(Xi) , with h ij in Z (-co, 00) ; the restriction of h ij to [0,00), and n(2,i,j) is the degree of the restriction of the =

ii) if nCI,i,j) is the degree

h

ij

of

func~ion gij(xi) = hij(-xi) to [0,"'), then n(l,i,j) - 2j and n(2,i,j) - 2j < n(Z,i,O) for all i and all j > 0;

°

iii) Pij ~ for alJ. i and j, PiO 2 PiN is non-vanishing for all i.

E

>

°




°

Let L

N iDi PiD i ,wlt · h eac h Pi LO(-l)

=

on the interval [a,oo).

~

0, Po

~

E

> 0,

Suppose each Pi is the restric-

tion to [a,m) of an element of eW(_oo,oo). Then L is said to be limit-N on [a,oo) if there exist exactly N linearly independent solutions to Lf =

° in

L [u,oo). 2

A parallel defi-

nition applies to L on (-oo,aJ. Remarks:

It is well-known that for any L in the above definition,

there exists at least N linearly independent L [a,oo) solutions to 2 = 0. The same result holds on (-ro,aJ. Hence L can fail to be

Lf

limit-N on (-oo,aJ or [a,oo) only by having N independent square-integrable solutions.

+

1 or more linearly

It is also well-known that

if b > a, L is limit-N on [a,oo) if and only if L is limit-N on [b,oo); if b < a, L is limit-N on (-oo,aJ if and only if L is limit-N on (-oo,bJ.

Finally, it is well-known that, if L

=

L~(-l)jDjPjDj,

with each Pj in eoo(_oo,m), Pj ~ 0, PN > 0, and Po ~ E > 0, and L is not limit-N on some interval [a,oo) or (-oo,aJ, there is a non-trivial f such that Lf

=

°

and f is in L 2 (-00,00).

Remark:

Recall that, as discussed in section 1 it is shown in a 6 Kauffman [5J that there exist L of the form L =_D 3 x a D3 + bx - , with a > 6 and b > 0, such that L is not limit-3 on [1,00). Remark:

If L is limit-N in the sense of our definition, it is not

hard to show that the deficiency indices of the minimal operator corresponding to L on [a,oo) are both equal to N, and conversely. Hence our definition is equivalent to the usual definition Theorem 2.10. N

..

(Kauffman, to appear). .

Suppose M

=

L~., with l

Mi = LO(-l)JDIPijDI' Suppose Pij(x) = hij(xi)' where h ij is in eOO(_oo,oo) for i > 1, and hI. is the restriction to [0,00) of an ele00 J ment of e (_00,00). Suppose that hlN is non-vanishing on [0,00), and h iN is non-vanishing on (_00,00) for i > 1. each i, and each Pij is non-negative.

Suppose PiO

~

E >

° for

Let R be as defined above.

Let L· = LNO(-l)jDjh .. Dj. Then, if R is essentially self-adjoint, Ll l lJ is limit-N on [0,00), and Li is limit-N on (-00,0] and [0,00) for i > 1. Remark:

Although the examples given above are of Li which are not

limit-N on [1,00), for N

=

3, it is easy to extend these expressions

to expressions on [0,00), which can not be limit-3 by the above remarks. Proof of Theorem 2.10:

It is well-known that if Li is not limit-N

on [0,00) or (-oo,OJ for some i > 1, there is a non-trivial solution

277

.4 IVnYL THEORY

to Li f = 0 such that f is in L2 (_00,00) .

If Ll is not limit-N on

[0,00), then there is a non-trivial f such that Llf

=

°

on [0,00) and

f is in L [0,00), and such that (f(0),f(l)(0), ... ,f(2N-l)(0» 2

is in

S.

Select any i

such that Li is not limit-N on some half-line, and let

f be the square-integrable solution constructed above. k = f(xi) for x in R+. Note that Mifi = 0.

Define

fi(x)

Pick any ¢ in C~(_oo,OO) such that ¢ is identically one on a neighbor-

°

hood of zero, and such that ¢(y) = for lyl ~ 1. Let 6 j (x) = ¢(x j ) k for j ~ 1, and for any x in R+. Let g TI j i 6j f i . k o o k Note that Mig = and g is in L 2 (R+). Note that g is in C (R+) and all partial derivatives of g are extendable to continuous functions k-l on {xl Xl 20}. Note that Tg(z) is in S at any point z of R

°

+

k For any j, Mjg is in L 2 (R+). Hence g is in the domain of Q, where is defined in Theorem 2.5. By Theorem 2.5, if R is essentially

self-adjoint,

R

Q.

=

in the domain of

Hence, if R is essentially self-adjoint, g is

R.

It is not hard to see, however, that for any g in the domcin of (Mig,g)

~

c(g,g).

self-adjoint.

3.

Q

R,

This is a contradiction, if R is essentially

The theorem is proved.

UNANSWERED QUESTIONS

In conclusion, it seems worthwhile to list some interesting problems which have not yet been solved.

2

2

Problem 1. Let L D P2D - DplD + PO' with each Pi ;:: 0, PN > 0, and p·O ~ E > 0 on [a,m). Suppose each Pi is the restriction to

Is L necessarily limit-2?

[a,m) of an element of C"'(-oo,oo).

(Equivalently, is problem P well-posed for L?) Problem 2.

Let L

N

"

.

;:: 0, PN > 0, and j Po ;:: c > 0 on [a,"'). Suppose each Pj is the restriction to [a,"') of an element of CeD(_ro,w). Is it possible for all solutions to Lf = 0 =

LO(-l)JDJPjDJ, where each P

to be in L 2 [a,oo)? Problem 3.

Let M and Li be as in Theorem 2.10.

limit-N on each half-line.

Suppose each Li is

Is R necessarily essentially self-

adjoint? Problem 4.

Let M be as in section 2.

Can there exist two N2N dimensional subspaces Sl and S2 of complex 2N-space K ,such that

Sl and S2 are as discussed in the definition of problem P, and such

278

R.M. KAUFMANN

that problem P is well posed for Sl and not for S2? REFERENCES [1] Devinatz, A., Positive definite fourth order differential operators, J. London Math. Soc.

(2) 6 (1973), 412-16.

[2] Eastham, M.S.P., The limit-2 case of fourth order differential equations, Quart. J. Math.

Oxford (2) 22 (1971), 131-34.

[3] Everitt, W.N., Some positive definite differential operators, J. London Math. Soc.

(1) 43 (1968), 465-73.

[4] Hinton, D.B., Limit-point criteria for differential equations, Canad. J. Math. 24 (1972), 293-305. [5] Kauffman, R.M., On the limit-n classification of ordinary differential operators with positive coefficients, Proc. London Math. Soc.

(3)

35 (1977), 496-526.

[6] Kauffman, R.M., On the limit-n classification of ordinary differential operators with positive coefficients (II), Proc. London Math. Soc. (3) 41 (1980), 499-515. [7] Walker, P.W., Deficiency indices of fourth-order singular differential operators, J. Diff. Eq. 9 (1971), 133-41.

Spectral Theory of Differential Operators I.W. Knowles and R. T. Lewis (eds.) ©North-Hol/and Publishing Company, 1981

ON THE CORRECTNESS OF BOUNDARY CONDITIONS FOR CERTAIN LINEAR DIFFERENTIAL OPERATORS Ian Knowl es Department of Mathematics University of Alabama in Birmingham Birmingham, AL 35294 David Race Department of Mathematics University of the Witwatersrand Johannesburg 2001 South Africa

For ordinary linear differential expressions T of order 2n defined on a real interval 1, the problem of determining which linear homogeneous boundary conditions give rise to well-posed differential operators in L2 (I) is considered. For the case I = [0,00), it is shown that all the operators obtained by imposing n linearly independent (complex) boundary conditions at 0 are well-posed, under appropriate conditions on the coefficients of T. The regular case, 1= [O,lJ, is also discussed. The problem of correctly assigning boundary conditions to formal differential expressions arising from physical models, and elsewhere, is of central importance in applications of differential operator theory. For definiteness, consider the differential expression T defined by TY(X) = (_l)n y(2n) + nil (Pn_r(x)y(r)) (r), X E I, (1 ) r=O where I c R and the coefficients Pi('), 1 ~ i ~ n, are complex-valued and locally Lebesque integrable on I, We associate with T the usual maximal and minimal operators, Tl and TO respectively, in L2(I) as follows (see [12J): Let f[iJ denote the ith quasi-derivative of a function f (see [12, p. 49J). The operator Tl is then given by D(Tl)={fE L2(I): f[i], 0 ~ i ::c 2n-1, are locally absolutely continuous, and Tf E L2(I)} Tlf = Tf, f E D(T l ), while TO is defined to be the closure of the operator TO given by 279

IAN W. KNOWLES and D. RACE

280

OtTO)

{f EO(T l ): f vanishes outside some compact interval [a,S] TOf = d,

f

E

C

(1)}

OtTO)'

The essence of the problem of assigning boundary conditions is roughly the following: one must choose the boundary conditions so that the associated restriction, T, of Tl has domain optimally large in some suitable sense. In the best of cases this means that the spectrum of the operator T allows something like an eigenfunction expansion theory. If there are too few boundary conditions, one can expect the point spectrum of T to fill out the complex plane; and if there are too many boundary conditions the point spectrum of the adjoint operator may do likewise (in which case the residual spectrum of T may cover the complex plane). Clearly, a minimum requirement on the spectrum of T is that the resolvent set, p(T), be non-empty. The following necessary condition for this to occur forms a convenient starting point for our discussion: Lemma [2, p. 1311]. Let T be an operator obtained from T by imposing a (possibly empty) set of boundary conditions on O(T l ), and let A E p(T). Then the number of linearly independent boundary conditions defining T is equal to the number of linearly independent solutions of the equation TY = AY that belong to L2(I). Our main concern here is to investigate the converse result; i.e., to determine conditions under which an operator T, obtained from T by imposing the number of boundary conditions specified in the theorem, has non-empty resolvent. That the converse is not true in general may be seen from the following example: Let T be defined by OtT) = {f E L2[0,1]: f' is absolutely continuous, f" E L2[0,1], and f(O) + f(l) = f'(O) - fl(l) = O} Tf(x) = -f"(x), 02- x 2- 1, f E otT). Here, the associated characteristic polynomial for the eigenvalues of T is identically zero, giving Po(T) = [, where Po(T) denotes the point spectrum of T. Clearly it is of interest to document precisely when such pathological cases occur, as one would expect, among other things, that any attempt at a numerical solution of a boundary value problem involving such an operator, would fail. On the other hand, it should be noted that even if one knows that p(T) is not empty, the associated spectral theory can still be extremely complicated ([1l,§5.4; 10;

9]). In the sequel, we denote the regularity field of an operator T, TO eTc Tl , by n(T); the essential spectrum of T is denoted by Eo(T), and the residual spectrum by Ru(T) (c.f. [8,§2]). An extension T of TO is called well-posed if n(T) is not empty. It is known (see [8,§3]) that T is non-well-posed if and only if n(T O) C PutT).

281

BOUND,4R Y CONDITIONS FOR DUTERENTIAL OPERA TORS

We consider firstly the so-called regular case in which we take I ~ [O,lJ for simplicity, and assume Pi(·) EO L[O,lJ, 1 ~ i 2. n. Notice that n(TO) = [ , and thus an extension T of TO is non-well-posed if and only if Po(T) = [. Given matrices A = (a rs ) and B = (b rs ) of order 2n and with complex entries, define the operator TAB by V(T AB ) = {f EO L2[0,lJ: fCiJ, 0 < i < 2n-l, are locally absolutely conti~uou~, Tf EO L2[O,lJ, and

:~~

a rs f[s-lJ (0) + brs f[s-l] (1)

=

0 for

~r~

2n} (2)

For general n, rather little is known about which extensions TAB are well-posed. One can reduce the problem to the case p.1 (.) = 0, 1 -< i -< n, by means of known asymptotic formulae for the solutions y(X,A) of (T - A)y = 0 valid for fixed x and IAI ~ 00 (c.f. [11,12J). For separated boundary conditions the extensions TAB are always well-posed ([11, Lemma 3, p. 94J). For n = 1 it is not difficult to show that TAB is non-well-posed if and only if a12

a 21

b ll b 21

all

a12

a 21

a 22

all

+

a 22

b12 b22

0,

bll b 21

b 12 b22

0,

and

= o. In particular it follows directly that TAB is well-posed whenever the boundary conditions are J-selfadjoint (where J denotes complex conjugation in L2[0,1]; see It seems unlikely that for general n all J-selfadjoint operators TAB [8,13J). are well-posed, although there are no examples confirming this, as yet. n

x

We now concentrate on the singular case, I = [0,00). Let A = (a rs ) be an 2n matrix of complex numbers with rank n. Define the extension TA of TO by V(T ) A

~

{f

E

D(T ): 2f a f[s-lJ (0) = 0, l s=l rs

r=1,2,···,n}

(3) TA f = Tf, f EO V(T A) Then we have Theorem 1. If Pi = 0, 1 < i 2. n, then for every choice of the boundary matrix A, the extension TA of TO is well-posed.

Proof. Observe that by [5, p. 106], we have Eo(Ta) = [0,00), and hence that rr(T O) = C - [0,00); it is thus sufficient to determine when a complex number

282

IAN W. KNOWLHS and D. RACE

A ¢. a: - [0,00) lies in Po(TA)'

For such a A consider, then, the equation, (- 1 )n f (2n) = Af.

(4 )

Let p denote the 2nth root of (-1 )nA satisfying TI/2 < arg p < n/2 + n/n. Then . 2n, were h the distinct 2n th roots of (-1 )n A are given by lJi =- PE i - 1 , 1 :::.- 1:::'£ = exp(iTI/n). Any eigenfunction of TA must be of the form f(x) = c exp(px) + c 2 exp(P£x) + ... + c n exp(P£ n-l xl (5) l for appropriate constants c ,,·· ,c ' Using (5), one can show that the characterl n istic equation for the eigenvalues of TA has the form 6(A) = det(M) = 0, where M = (m ij ) and ((j-l))S-l _ 2n I a is p£ mij(p ) - s=l . -1 i-l We can write M = AG, where G = (g .. ) the 2n x n matrix with g .. = ( PE J ) in th . th 1J 1J the i row and j column. In this case the formula for the determinant of the product is

where A denotes the n x n matrix consisting of columns sl ,s2,···,sn of A, sl ... sn and Gsl ... sn denotes the n x n matrix consisting of rows sl ,s2"" ,sn of G. The equat10n for the eigenvalues of TA thus becomes sl+s2+···+ s n- n si- l

I

P

( IT

1':-:h<s2j

Clearly, TA is well-posed if and only if equation (7) is not identically zero. As A is of rank n, at least one of the determinants det As ... s is not zero. 1

n

: det A t O}. Then, by [11, Lemma 2, p. 91] n sl' .. sn the term in ps-n in (7) is non-trivial, and the result follows.

Let s

=

max{sl + s2 + ... + S

#

Remark. This result is of independent interest. In much of the qualitative spectral theory of non-selfadjoint differential operators, one is forced (at least implicitly) to make artificial assumptions in order to exclude the "bad" extensions (see e.g. [1, p. 11, 9,. 15; 4; 7, §9]). It is therefore very useful to know precisely when such extensions cannot occur. Provided the coefficients Pi' 1 :::.- i :::.- n, are not too large, Theorem 1 can be extended to cover more general operators c. More precisely we have Theorem 2. If Pi = qi + r i , where qi E Lm[O,m), and r i E L[O,m), 1 :::.- i ~ n, then the extension TA of To defined by (3) is well-posed for every choice of the boundary matrix A.

m.

Remark. This includes the case Pi E L 1[0,00), 1 < m < 00, 1 ~ i ~ n, as one can i always write an r-integrable function, 1 < r < 00, as the sum of an integrable function and a bounded function.

283

IlOUNDAR Y COIVDITlONS FOR U1I'H'RENnAL OPERATORS

Proof.

The proof is divided into several stages.

the equation TY

Following [12,§22.2] we write

J..y in system form as

=

dY = A(x) Y dx (y,y[l], ... ,y[2n-1 J )T, and A(x)

=

AO(x) + Al(x) where AO and

Al are defined in [12, p. 176J, and we have set Po

=

1.

where Y = Y(x,r)

=

(8)

Let

Y = BU

( 9)

where

B

III

1l2n

n III

n 1l2n

n+l -Ill

-11

(_ 1 ) n- lll~n- 1

(10)

n+l 2n

(_ 1 ) n- \~~- 1

and Ili' 1.::. i.::. 2n, are the distinct 2nth roots of (-l)nJ,. defined earlier. Set B- 1 = (B ij ) and define A2 and A3 to be the matrices obtained from Al by replacing the elements p,., 1 < i ~ n, by q. and r., respectively. The system (8) then

,

,

becomes (11 )

(12 )

It is not hard to see that the functions c ij (x,·) and f ij (x,·) are analytic for fixed x> O. Also for Ipl ?:- 1 we have ICij(x,p)1 ~Kn

n

I

k=l

(13)

Iqk(x)l; Ifij(x,p)I.::.L n

where Kn and Ln are independent of x and p. Our initial goal is to determine the asymptotics of certain solutions of (11), from which we easily obtain the behaviour of solutions of TY = J..Y via (8). Before doing this we digress for a moment. H

=

C-

(f

x

o and consider, for


R). Let V be a real 1 1 0 valued function on Q and let H be any self adjoint realization of -6 + V where n 6 denotes the n dimensional Laplacian. From time to time we will also let n r = 11";(11. II.

THE RESULTS OF KATO AND AGMON.

I r V (x) I

(a)


R 0

then the equation Hu had no nontrivial L2('l) solution if

Ie >

= AU

K2.

On the other hand, Agmon ([1 J, Theorem 4) showed the following for V E L~oC(rl) : If (i ).

V is locally Holder continuous in a connected open set has measure zero and Qo :J {x I I xii > Ro} and

Q

o

C ~,

Q -

Q

0

(ii). For r .:: Ro ' V(x) Vo(x) + Vl (x) where Vo(x) is continuous and has a continuous radial derivative such that 295

296

HOWARD A. LEVINE

(r -)- +00) (b) (c)

lim sup r dV lor = ~ r -)- +00 0 0 1 V1 (x) = 0 (r - - d (r -+ +00,

some

then H has no eigenvalues where A >

€

> 0),

~o/Z.

The principal result we set forth here includes both of these results as special cases. III. THE RESULTS OF KHOSROVSHAHI, LEVINE AND PAYNE. the following result (Theorem 4 of [4J).

In [4J we established

Suppose V E L1oc(Q) and is real valued. Assume that V satisfies (i) of Agmon's result and that for some R0 sufficiently large and all x, /lx/l > R , - 0 (iil'. V(x) = Vo(x) + V (x) + VZ(x) where Vo(x) is real, continuous, l possesses a continuous radial derivative and (a), (b) above hold, V (x) is real l and satisfies (a) above, and VZ(x) is such that (writing ~ = r~) p

(d)

sup f oV(o~) do ~ M < % for all Iltll=l r

p,

r ~ Ro .

Let a =

max

I I

{K + [K 2 +

2~o(1-ZM)2ii}Z

2K2 +

~0(1-4M)

L

4(1 - ZM)Z 2(1 - 4M)Z 2 Then H has no L (Q) eigenvunction corresponding to A if A > a.

l J

Notice that if V = Vo(K = M = 0) we recover Agmon's result, while if Vl (110 = M = 0) we obtain Kato's result. Observe also that condition (d) can hold for potentials Vz which are not solely functions of the radial variable. V

IV.

THE

Let u E

~lAIN

THEOREM.

In [4J we establ ished the following result.

2,00(Q) n L2(Q) be ~ solution of W f1 u

n

where p(x)

E

=0

L1oc 2 (Q), ~ real valued and can be written, for Ilxll

for some sufficiently large R*. (A)

+ p(x) u

sup Ilxll>R*

Irpl(x)I~K

Suppose that

>

R* > R0as -

297

POSITIVE SPECTRUM OF SCHROHDlNGHR OPERATORS

(8)

sup

Iltll=l

If

p

ap2(a~)dal ~ M < ~,

p,

r .:: R*

r

(C)

po (x) is areal with a continuous radial - continuous function -- - derivative such that -po (x) -> Kl > O.

(D)

r apolar + (2 -

(E)

y -

4M - [1 )po(x) - K2/~




O.

2 r aPolar + 2(1 - 4M - £3)Po(x) - 2K /(1 - 4M - £4) .:: £5 where the £i's tend to zero ~ R* + +00. Then.!i pix) i2- Holder continuous on ~ connected open subset of r, r 0 say, and meas (,' - [Jo) = 0, it follows that u = O.

If we identify pix) with A - V(x) so that po(x) = A - Vo(x), Pl (x) = -Vl (x), P2(x) = -V 2 (x), then as r + +00, po(x) ~ A, r aPolar + - Ao. Conditions (O,E) then yield (at r = 00) (D' )

- 1\

(E' )

- 1\

o o

+ A (2 - y +

4M) - K2 /y

>

0,

2>-(1 - 4M) - 2K 2/(1 - 4M)

>

o.

The conclusion of Section III then follows after choosing the optimal

y

in

(0' ).

In order to establish the Main Theorem we make use of the following Lemma which has appeared in different forms in various places [1,3J. Lemma I. Let F(t) be ~ nonnegative function on (O,toJ, continuous there and twice continuously differentiable on (O,t ). Let c ' a , a , £ be constants with o l l 2 c l > 0, £ > 0, a2 > 1 and al + a2 > 1 + 2c l . If (*)

t

fOn

o

-a2

F(n)dn R*,

where 2C = y + n - 2 (1 - M) + E for some y > 0 and where small ~ one pleases if. R* h sufficiently large.

E

>

0 ~ be made ~

Lemma II may be established by means of a Rellich type identity as follows: R

J

r

~ (pu p + CU)(lIu + pu) ds dp = O. Sp

An integration by parts followed by the use of the inequalities 81, 82, C1, C2, C3 of the Appendix then gives the Lemma. Finally one employs Lemma III.

R

Let u, p be

~i!!

Lemma II.

Then

'"

J 1 J P \grad u\2 ds do dp

r

p

p

S0

~lz(l+d

J r-

S

o

12 u ds + (1+E)

J -1 J P (po+du 00

r

00

p p

S

0

2

ds do dp.

299

POSI1'lVE SPECTRUM Of' SCHIWEDINGLR OPl'RArORS

These are combined to yield the desired lower bound for FF" - (F,)2 (see[4J). The computations are very tedious. V.

SOME EXAMPLES AND REMARKS.

Example 1.

Let V(r)

Then if we take V(r)

=

Ar~ sin(r B).

V2 (r), condition (d) of Section III becomes IA JP 06+1 sin(00)dol 2

~A r 2+6- S

r

So if 2 + 6 < S, we may take M = (2A/S)(R*)2+6-S as small as we please if R* is sufficiently large. Thus -6 + V has no eigenvalues \ E (0,00). Example 2.

Let V(r)

where A, B are positive constants. von Neumann and Wigner.

=

~r sin (Br)

This includes the class of potentials of

We find that if we take

(i)

V

V2 ' then there are no eigenvalues in (0,00) if 2A/B

(ii)

V

Vl ' there are no eigenvalues in (A 2 ,oo) if y. ~ 2A/B

(iii)

V

Yo' there are no eigenvalues in (izAB,oo) if ZA/B .:: l.


r > R* > Ro where R* is taken large. R

\ J ¢ r

Sp

R 2 PPPluudsdp)

"

by the Cauchy-Schwarz inequal ity.

"

The conclusion now follows.

(4)

(5)

306

ROGER T. LEII'IS

The lemma obviously holds for all ~ E C~(Q), which will later allow us to apply it to the case of the Dirichlet problem. However, for some sets Q and certain choices of functions g(x), a much wider range of applications is possible. Since ylg = '7g.", where" denotes the unit outward normal at xEr, then '7g'" ~ 0, for each x E r, will insure that inequality (5) holds for all = n ~ E Co(lR ). Example 1. Suppose that, for some a E Cl(Rl) and r = lxi, g(x) = air) for all x E r. Assume that either r does not contain the origin or that a' (0) = O. Then, the inequality a' (r)(x l ,'" ,x n ) . ,,~ 0, X E r, (6) implies that (4) and (5) hold for all ~ E C~(lRn) provided "'ng > 0 on Q. Example 1 follows from the fact that Ylg any nonzero x E r.

=

I7g'"

=

r-la'(r)(xl,···,x )·" for n

In some of our appl ications, when g is a radial function as above, a' (r) will be nonnegative. In order to insure that inequality (5) holds for all ~ E C=(lRn) we will need to have r = r- u r O where r- and r O are defined as o follows: . + r O, an d r Defl-ne r,

to be the set of all x E r such that

(xl" .. ,x n )· ("1'" . '''n) is positive, zero, or negative, respectively, where" is the unit outward normal vector at

= ("1 ,'"

'''n)

x E r.

Note that x is in r+, r O , or r- according to whether the angle 8 between the point vector from the origin to x and the outward normal vector" is acute, right, or obtuse. For example, if Q is the exterior of a ball that is centered at the origin, then e = ~ and r = r . Corollary to Lemma 2. 4

J

If

~

E

C1 (n '" {O}), then o

Ixl S 1'7~(x)12 dx.:: (s - 2 + n)2

J

IxI S-

2

1~(x)12 dx.

n Q Mar'eover, for n > 2, inequality (7) is valid for all 1 n + ~ E {u E Co (lR '" {O}): u (x) = 0 on r } when B > 2 - n and when S

valid for aU

~

E

{u E

C~(lRn


0 such that for all X E Q 00

1 sl-n

~A(s)-ldS

r

r

1 sn-l

5 co w(s)[J t n- 1 w(t)dt f t l - n ~A(t)-ldt]-l

ds
-'" i3

xl

tn-lw(t)dt

lim {w(xrlq(x) + [f tn-lw(t)dt Ix 1-+ Ix I respectively. In the special case of \1 = R n and w(x) 00

(tl-nlJA(trl dtrl) = '" Ixl Ix I

f tl-nlJA(trl dtrl}= "',

(14)

(15)

(J

= 1, a theorem of Schechter [16, p. 192] (ef. Lemma 3.3 of [17J) can be used (see [llJ) to remove condition (10) of Theorem 1. Consequently, we can conclude from the Corollary to Theorem 4 below that when Q = JRn, q(x) = 0, w(x) a 1, and min e.v. A(x) = max. e.v. A(x)= )lA(lxl), then

311

SPECTRA OF SOM}, SINGULA /( ELLIPTIC OPhRA TORS OF SECO.'VD ORDER

lim Ix 1-+

Ixl n (l-n ()-l dr Ixl r iJA r

00

=

0

is necessary and sufficient in order that the spectrum of Th be discrete. Secondly, we illustrate the method for the case in which Q is bounded and the singularities of s occur on a portion of r. Assume that Q = Qn-l x G where n l Qn-l ~ R - and G C (O,p] for some p > 1. Assume that the singularities of S occur onl y on r n {x E Rn : x = O}. (We refer the reader to the book of n t~ikhl in [12, p. 207] where M. M. Smirnov has considered a similar problem.) Let w be a function of one variable such that w(x) ~ w(x n ) for x E Q. Let iJ A be a nonnegative function of a single variable satisfying min e.v. A(x) -> iJA(X n ) for x E Q. Suppose that for each k there is a ck' such that iJA(x n ) ,::c k > 0 for x E Qk. Note that the case in which iJA(O) = 0 is included. We would need only to choose each Qk in order that fk does not intersect the plane xn = O. For example, this problem appears to be associated with the study of heat flow along a rod with an end (x n = 0) which is completely insulated (see Mikhlin [12, p. 156]). The case in which iJA(x n ) -+ as xn -+ 0 is also included. 00

Theorem 3. Let

f

Assume that u(s)

x

n iJjil(S)dS

p

ff

w(t) (

xn

o \7..

f

1

t

= 0 on

r- u {x E r: X n t 1

f

w(x) dx l ·· ·dx n

Qn- 1

= O} for aZl

IJ; (s)ds)-

1

U E

O(S).

dt be bounded on

a

If

-1

t

p

lim f f w(x) dxl···dx n f IJ A (s)ds t-+O+ t Qn-l a

0

then Th has a discrete spectrum. 1 t -1 )-1 Proof. Let g(x) = f (f IJ A (s)ds dt and

xn

1

h(x)

=

f f

1

0

p

w(s) (

xn t

ff s Qn-l

w(x) dx l ·· ·dx n

f

s

_1 -1 IJ A (v)dv) ds dt.

a

The proof now

follows the proof of Theorem 1. Theorem 4. Let

f

p

Assume that u(s)

xn

1

iJji (s)ds J

xn bounded on

Q.

=0

on f

+

{x

E f:

xn = o} for all u

( ) dx 1 ·· .dx n f IJA-1 (s)ds )-1 dt (f f x w() tw a Qn- 1 t

If

t

w(x) dx 1 ·· .dx n

o Qn- 1 then Th has a discrete spectrum.

Let g(x) =

f

1

xn

E

O(S).

p

f f Proof.

U

t

(f t

p

1

iJji (s)dsr

1

f

p

1

iJA (s)ds

t

dt and

o

(s

>

0) be

ROGER T. LEWIS

312

s

t

J

h(x)

w(s)

s

3.

( JJ 0

Qn-l

p

w(x) dx l " 'dx n

J

-1 ~A

(v ) dv )-1 ds dt.

s

NECESSARY CONDITIONS FOR DISCRETENESS OF THE SPECTRUM OF SECOND ORDER ELLIPTIC DIFFERENTIAL OPERATORS

In this section, we show that the theorems of the last section are sharp, at least in certain cases. Our main device for doing this will be the following theorem that can be found in the book of Glazman [5 , p. 15]. Theorem 5.

A necessary and sufficient condition for the nwnbe" of points of

A, lying to the left of a given point Ao'

the spectrwn of a self-adjoint operator

to be an infinite set, is that theroe exists an infinite dimensional set G C D(A) for which

(Au - Aou, u) < 0 for all U E G. Since -6 of section 2 is symmetric then This self-adjoint [9, p. 323] and Theorem 5 applies to Th. Corollary to Theorem 5.

If there is an infinite dimensional set M C D(I1)

such that

h(u, u)


1

o -1/2

-

b

(f YA1(s)dSr l




B + E/2


0 ,

¢, f E A,

.e

¢(r,e) = Pr(z) * f(z) , z = re 1

,¢(r,o)

1:

n=O

a

~

n n

(r,e)

is clearly one-one and by Hopf's maximum principle, it is uniformly convergent on compacta of D. Hopf's maximum principle also assures that 2 Re Po(z) ~ c>O, (o,z) EO. Therefore, by Korovkin's theorem [9]on positive

321

RECAPTURING SOL( TIONS OF .4N ELLIPTIC EQl'A'FJON

operators,the eqn. (5) extends continuously to the dO since the associate f does. The map TC sends the class A one-one into the class R. Let f sA, then by the above ep = TC(f) E R = rUiB (0) because s ie i8 ep(l,e) = fee ), was established by the iden~ity ~ (~,e) = ~ (e ), n > O. More.e n n over, if ep E Rand feel ) = ep(l,o) , then the results of [5] apply to cons truc t E a (unique) f E A for which 1> TC(f). Indeed, the linear spaces A and Bare E E E isomorphic under the map TC' Conti nui ng . in thi s di recti on, we defi ne the sup-norm I I9 II r = sup {lg(pe1G)I:(l E RE and f = TC- (ep), II f ll r .:: Ilepll ' r < 1 so that 111>11 = Ilfll. To summarize, Theorem 1. The linear space ArlB E of analytic functions and the linear space Rn BE of regular solutions of L(ep) = 0 are isometrically isomorphic for each fixed E > O. THE MEAN BOUNDARY VALUES Approximate solutions with error bounds are constructed from smooth data at equally spaced points on the aD. The construction extends to data at points along a subarc of the aD at the loss of the el'ror estimates. Conformal equivalents and an interpolation problem are investigated. The constructions focus on the arithmetic means n

0n(g;8 1 ,8 2 ) = lin

L

k=l

g(exp (i21Tk(8 Z-8 1 lin + i21T8 1 ))

and

n=1,2, ... of a continuous function g on an arc {e 21Tis ; e .::s.::s 2}. l 0n(g;O,l) ~n(g;O,l), n = 1,2, ... , o,,,(g;o,1) = lim 0n(g;O,l).

Note tha t The shifted

n->=

means of g, . n-l \!n(g;O,o) = g(e 21TOl )/2n + lin L g(exp(i21T(2k-l)6/2n-l), k=l 21Ti8 n=1,2, ... app 1y to a proper subarc {e ;0'::S.::6}, 0O,

p

0

(¢) =

C5

, 00

represents q, uniformly on the cl (D). If C5 n(¢;O,l) = 0 for all n Furthermore, the following estimates are uniformly valid in e k 0 I¢ (r, 8) - 2: p (rjJ) 'JI (r, 8 )1< K( 6, iJl ) k(8) n=O n n for all k

~

1, 6
1 and r < 1.

Having established the basic representation theorem, we direct our attention to the interpolation problem. This is another consequence of the extension of the series representation of 'C(f) to the dD. Theorem 3. Let {a } and {B } be sequences of real numbers that converge to a n n 3+ 2+E) and 0 respectively, with the rates an-a = O(l/n E) and Bn = O(l/n for some E > O. Then there exists a unique function iJl E R + E ,for some E' > 0 such that: 2 (i) L(¢) = 0 in D and

for all n=l ,2, ... (10)

Furthermore, the series n=l (an-a)An(r,e) + n=l Bnrn(r,e) + a

converge uniformly to q,(r,e) on the cl (D).

Here, the basis is

RliCAPTURlNG SOH'nONS OF AN liLLlPllC EQUA nON

(~n(r,e)

+

(~n(r,e)

~n(r,e)}

323

/2

'n(r,e)} /2i

for all n=1,2, .... Proof.

From the sequences {an} and {Sn}' construct the functions h(z) =

(11 )

(a

L

n=O

-a)~

n

n

(z), k(z) =

L

n=O

S ~ (z) . n n

These are analytic [see 5] so the representations H(r,e) = Te(h(z)), K(r,e) = Te(k(z)) are uniformly convergent in the cl (0) and L(H) = L(K) = 0 in O. Because d2/1e2~ (r,e)=d2/a82~ (r,_e)=a2/a82~ (r,e),s>O , ss s the conjugate functions H = Te(h) and K = 'e(k) are solutions of L = 0 where 'e(~) refers to the expansions in eqn. (11) conjugated. Oefine the following functions: U(r,e)

=

[H(r,e) + H(r,e)]/2

V(r,e) = [K(r,e)

'e((h+h)/2)

K(r,e)]/2i

If 1> = U+iV = 'e(f) , f = (h+h)/2 + (k-k)/2i, then L( 0). 1 1 1 1 Suppose now we change the integral condition (8) (or (7) as follows. Let zA be solution of z(4) - AZ = 0 which also satisfies z(o) = z(l) (0) = z(l) (1) = 0, z(2) (0) = 1. We then require that there exists Rl(s.t) £ that functions (17)

ZA (s)

+ fS Rl (s ,t)

ZA (t) d t

o

,

e[o

< t < s < 1] such

A >

°

satisfy a fourth order equation (18)

B\ _ Ay 0, 0 ~ s ~ 1, eigenvalues, Y ' Y*, i = 2,3 •... A A

y~4) + (Aly(l)) (1) +

=

and that 71 ,71 * ,71 * , ... are are eigenfunctions, 1 2 3 . . . * * 1 i . wlth normallzatlon constants, P ,P2 P3' ... for the elgenvalue problem consisting l of the above differential equation and boundary conditions,

332

JO YCli McLA UGHLIN

~ yell (0) ~ yell ~ yell (1) ~ a

yeo)

(19)

What has been done then is only to change the set of solutions of z(4) - AZ in the integral relationship used to define solutions of y(4) + (Aly(l»

i'?y - AY ~

~ a

(1) +

O.

l The same proofs, used to obtain Theorem 1 can be employed to determine Rl, A , and Bl uniquely.

a

< s < 1.

Furthermore, it is not true that Al

= Al

and Bl

The proof of this is by contradiction as follows.

= Al, Bl = Bl, for y, = YA y,* = Y'*' Al l' Al Ai

Al

(8) and (17)

a i

< s < 1, then Rl _ Kl for

a

2,3,4, ... ,

< s < 1.

a

< t

= Bl

for

If

< S < 1 and

The two integral relationships

imply that

a

+ fS K(s,t)

a

* i ~ 2,3, . . . . The theory of Volterra integral equations yields A , Ai' l This last equation is false and the desired contra(s) - ZA(S), a ~ s < 1.

for A

zA

diction is obtained, when A ~ A . l section 2: In this section we will present a continuity result for solutions of fourth order inverse eigenvalue problems. The result which will be presented shows that A and B vary continuously as the A.' sand P.' s vary continuously from the 's and P ~'s. 1

t.

~

1

1

More particularly a bound can be determined on the L'" norm of A, A(l), and B in

*

1

terms of the differences Ai - Ai and

Pi sufficiently small.

1

*

as long as the differences are

Pi

The theorem will be presented along with a brief explanation of the proof. Theorem 3:

* p., * i ~ 1,2, ... be defined as before. Let z;\, A.,

Let

1 1 1

AEC

[0,1], BlOC [0,1].

constants for (3).

Let Ai' Pi' i ~ 1,2, ... be eigenvalues and normalization

Let YA satisfy the differential equation of (3) for

°
0 such that for A. ,A.* > QM K are determined by equations such as l 1

bounds for individual terms in

fl

o -B(~) (YA.-YA~)dt]dt, l

and Y . (s) = zA. (s) A 1 1 If either A: or Ai

:5..

+

f 0

1

f(s,t,\) [-(A(t)y . A l

(1)

)

(1)

1

- By . (t)] dt. A 1

Q M, bounds for the individual terms in K are determined by

equations such as

[(\)~

-

(Ai)"']Y(S,Aj') + (Aj'-\) fSy(s-t,Ai)y . (t)dt A

o

1

and (1) (t)) (1) - By . (t) ]dt. Ai Al Combining the resultant bounds yields (18). Y . (s)

A

= zA(s)

l

Section examine section inverse

l

+ fly(s-t,A.) [- (AY

o

l

3: We would now like to return to the integral assumption (8) (or (7)) and it more closely. As a reminder, we have already shown, in the example of 1, that a change in this assumption produces a different solution of the eigenvalue problem.

What we observe as a result of these assumptions can be described intuitively as follows. The assumption of the integral relation (8) implies that the spectral data for an associated non-self-adjoint eigenvalue problem is the same when the 4 1 1 f' .. dl. ferentlal equat10n 1S . e1 t h er Z (4), -AZ = 0 or Z ( ) + ( Az ( ) ) ( ) + Bz - ' AZ0 = • To be more specific, let us determine the solution of the inverse eigenvalue problem given by Theorems 1 and 2 of Section 1. Further, let P. ,~"!" i = 1,2, ... be l

l

the normalization constants and eigenvalues (with associated eigenfunctions zi (s)) for the eigenvalue problem (19)

z(4) - AZ = 0, z(O) = z(l) (0)

=

z(2) (0) = 0 = z(l).

z.

The eigenfunctions zi (s) are normalized so that (3) (0) = 1, the adjoint eigen2 functions are normalized so that (1) (0) = 0 alnci p. is the L inner product of l.,a l,a 1

z.

z.

334

Zi

JOYCE McLAUGHLIN

Z.l.a

and

(4)

Y

(20)

Then the eigenvalues and normalization constants for

+ (Ay(l»

(1) + By _ AY

= o.

y(O)

=

y(l) (0)

=

y(2) (0)

are also Ai' Pi' i = 1.2 •... where the associated eigenfunctions ized by y(3) (0) = 1, and the adjoint eigenfunctions 9(1) (0)

y.l,a

=

y(l)

=

O.

YA

are normali are normalized by

= 1.

Finally, it should be noted that. using the techniques developed by Leibenzon [12]. it can be shown that there is exactly one pair of coefficients A(s) € cl[o.ll and B(S) € C [O,lJ such that the eigenvalue problems (3) and (20) have eigenvalues and normalization constants Ai' Pi' i = 1.2 •... and

~., l

P.• l

i = 1.2 •... respectively.

Hence Theorems 1 and 2 produce this set of

unique coefficients. REFERENCES

[1]

V. Barcilon. Iterative Solution of the Inverse Sturm-Liouville Problem. J. Math. Phys .• 15 (1974), pp. 287-298.

[2]

V. Barcilon. on the solution of inverse eigenvalue problems of high orders. Geophys. J. R. Astr. Soc .• 39 (1974). pp. 143-154.

[3]

V. Barcilon. on the uniqueness of inverse eigenvalue problems. Ibid:. 38 (1974). pp. 287-298.

[4]

G. Borg. Eine Umkerung der Sturm-Liol1villeschen Eigenvertaufgabe. Acta. Math .• 78 (1946). pp. 1-96.

[5]

E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations. McGraw-Hill Book Co., New York, 1955.

[6]

I. M. Gel'fand and B. M. Levitan, On the Determination of a Differential Equation from its Spectral Function, Izv. Akad. Nauk SSSR Ser. Mat., 15 (1951), pp. 309-360; English transl., Amer. Math. Soc. Transl., 1 (1955), pp. 253-304.

[7]

o. H. Hald, The Inverse Sturm-Liouville Problem with Symmetric Potentials, Acta Math .• 141 (1978), pp. 263-291.

[8]

H. Hochstadt, The Inverse Sturm-Liouville Problem, Corom. Pure Appl. Math., 26 (1973), pp. 715-729.

[9]

M. G. Krein, On a Method of Effective Solution of a Inverse Boundary Problem, Dokl. Akad. Nauk SSSR. 94 (1954), pp. 987-990.

[10]

M. G. Krein. Solution of the Inverse Sturm-Lionville Problem, Ibid., 76 (1951), pp. 21-24.

[11]

Z. L. Leibenzon, The Inverse Problem of the Spectral Analysis of Ordinary Differential Operators of Higher Order, Trudy Moskov. Mat. Ob~~., 15 (1966) pp. 70-144; Trans. Moscow Math. Soc., 15 (1966) pp. 78-163.

[12]

N. Levinson, The Inverse sturm-Liouville Problem, Mat. Tidsskr. B., 25 (1949), pp. 25-30.

[13]

B. M. Levitan, Generalized Translation Operators and Some of Their Applications, Fizmatigz, Moscow, 1962; English trans. Israel Program for Scientific Translations, Jerusalem and Davey, New York, 1964.

[14]

B. M. Levitan, On the Determination of a Sturm-Liouville Equation by two Spectra, Izv. Akad., Nauk SSSR Ser. Mat., 38 (1964), pp. 63-78; Amer. Math. Soc. Transl., 68 (1968), pp. 1-20.

FOURTH URDU? INVf!RSE EIGENV,1LUE PROBLEMS

335

[15]

V. A. Marcenko, Concerning the Theory of a Differential Operator of the Second Order, Dakl. Akad. Nauk SSSR, 72 (1950), pp. 457-460.

[16]

J. 11cKenna, On the Lateral Vibration of Conical Bars, SIAM J. Appl. Math.,

[17]

J. R. McLaughlin, An Inverse Eigenvalue Problem of Order Four, SIAM J. Math. Anal., 7 (1976), pp. 646-661.

[18]

J. R. McLaughlin, An Inverse Eigenvalue Problem of Order Four - An Infinite Case, SIAM J. I·lath. Anal., 9 (1978), pp. 395-413.

21

(1971), pp. 265-278.

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Spectral Theory of Differential Operators I. W. Knowles and R. T. Lewis leds.} © North·Holland Publishing Company, 1981

STURM THEORY IN n-SPACE Angelo B. Mingarelli Department of Mathematics University of Ottawa Ottawa; Ontario Canada Dedicated to Professor F.V. Atkinson on the occasion of his sixtyfifth birthday. Two conjectures are formulated regarding a form of the Sturm Comparison Theorem for a second order vector differential equation. These results are verified in particular cases and it is noted that their validity would lead to a form of Sturm's theorem for both self-adjoint and non self-adjoint equations. I NTRODU CTI ON In 1930 Marston Morse [4J formulated a version of the Sturm comparison and separation theorems which, when applied to the vector equation y" + Q(t]y

o

(1.1)

where Q(t) = Q*(t), Y E R n , yielded a natural extension of the said theorems to this sett ing. Ilis results were extended by Hartman and Wintner [3J. Recently a version of Sturm's theorems was discovered by Ahmad and Lazer [lJ for non self-adjoint systems of the ahove type which do not, however, extend the results in the self-adjoint case. The purpose of this note is to present a version of Sturm's theorems which appears to include hoth the self-adjoint and non selfadjoint cases mentioned above. 2. We will assume hereafter that, unless otherwise specified, all matrices P(t), Q(t) are continuous nxn real valued matrix functions whose eigenvalues arc all real functions on I = [a,bJ The points a 7 S in I will be called (mutually) conjugate if there exists a non trivial solution of (1.1) such that y(a)=y(S) = 0 The equation (1.1) will be termed disconjugate on I if I fails to contain any conjugate points, ie. if every non trivial solution of (1.1) vanishes at most once in T. CONJECTURE 1. Let Al (t), A 2 (t), ... , An (t) denote the eigenvalues of QCt). If for each t ( I , maxOl (t), ... , An (t)} ,; 0, 337

(2. 1)

338

ANcrLO H. MINCARELLI

(1.1) is disconjugate on [a,b]. Since we assumed that the eigenvalues of Q(t) are all real, note that

where

Amax{Q(t)}

CONJECTURE Z.

is the largest eigenvalue of Q(t).

Let pet), QCt) be as above. y"

and let Z(t)

~

+

l'(t)y

=

Consider

0,

(Z . Z)

o

(Z . 3)

n be a solution of Z" + QCt)Z

satisfying Z (a)

II

Z (b)

( Z .4)

.

If ,\

max

{P(t)},,'\

-

max

{Q(t)}

( 2 . 5)

for each t ( I , equality not holding everywhere on T, there exists a solution yet) ~ 0 of [2.2) such that yCa) 3.

=

y(c)

=

n

a Ao }' The following statements are

.pon

345

SE LFADJOINTNLSS OF MATRIX OPliRATORS

equivalent : (i) !t(A- A) is dense for some A E C 0 (ii) ~(A- A) is dense for all A E C 0 (iii) 9.-(H - \q + A2) is dense for some \ E \ E (iv) ~(H - Aq + A2) is dense for all (v) H is essentially selfadjoint.

C C

0 0

We sketch the proof of Theorem 1b) and Theorem 2b) (the rest is rather standard). Proof of Theorem 1b} • It is easy to see that it is sufficient to consider positive definite H and set T = H. We prove only the statement concerning A; the other part is easier. By (QB) q can be extended to q E :£ ( ~11 ~ -1) • Define _ _ ":;D(A) ={U Ell VE;!t1 :vE~1 ' UEH,-l~v+JJ(H) Au Ell v_~lv Ell (-H'u+ qv)}. It is easy to see that A is injective, A E Je (';1(,1 ) • We have to prove that if H is essentially selfadjoint then A = A. _ Let UEil v E X) (A). Then v E~1 , U = U + E,-lqv where U Eel> (H). Choose vn E.p(q)n ifa,1 ' un E~(H) such that vn -----+ v in ~1 (H+1) un -..;. (H+l)u , in ~(we used (QB». Since ~(H) is dense in ~ we can find un E J:J (H) such that II-Ho. + qv I < _- 1 , --1 n n Set wn = H (-HUn +qv n ) -un + H qv n · Then wn ->- 0 , Hw - - + 0 in ,/, • Let un U + U • Obviously u E J) (H) and n a n n n

*.

A

1 im

n->-oo

U

n

in ~ hence find

=

1 im

n+oo

U

n

+ l'

A

n!: un - -1,

=

(note that H' q E __ -HUn = qV n --+ -Hu Hl / 2 + H,-lqv)

(_u n

1.

(

U - n~~

- -1

wn - H qv n )

=

-

- , -1,

U + H

qv

=

U

t ( "'1) 'J

) • Further HUn Hu - HW n + qVn' - 1/2 n in ~. From H w --+ 0 in ~ we = El/2~ + El/ 2 (H,-lq?v - H,-l CfI)-----+ 0 n

n

in ~ • Since un converges to U in ~1 we conclude that un converges to u + H,-lqv = u in ~1 • Thus we have found zn = un Ell vn E JJ (A) such that zn --+ u Ell v AZn ---+ v Ell - Hu = AZ in 'Jt 1 (i) , (ii) and (v) are equivalent by an eorem 2a) We shall prove that if A E Co then the range of A - \ is dense iff the range of H( A ) = H - A q + A2 is dense. This obviously implies the equivalence of (i) - (v) • Assume Z = x Ell Y 1.E.'9.. (A-A). By Theorem 1c) Co C p(1\.) (A as in the proof of Theorem 1b) ), therefore Z It(A- A). Let U E;D(H) = JJ (H(A)) then U Ell AU E!l(I) by Theorem 2a) and 0 = (zl(A-A)u Ell AU) = -(y/H(A)U), hence y is orthogonal to 5t(H(A)) . • Also 0 Ell v EJ)(I) for every VEJ) (E) and (zi(A-\)OEllv) =0 = (T 1 / 2 xI T1/2v) + (y(q-A)V) = (x + [(q_\)T-lj\\TV). We conclude

346

lJRANKO NAJMAN

YE ,1.(H(!..)) .1

-[(q- A)T -1 ]

X =

Conversely, if x, y satisfy (1.6), then orthogonal to ~ (A- A).

z

*y

=

x

( 1 •6)• 6)

Y E J(,1

is

As regards Theorem 1c) • note that if q is skew-selfadjoint, then -A, -B are similar to A' , B' ; A' • B' are A, B with q replaced by -q • Thus A, B generate Co-groups. 2. Application We apply the preceding results to the case ~= L2(Rm) • , H is the Schrodinger operator, q a multipli~ation operator. 2

m

Set

I

(-io.-a.) j=l J J

L=

, where

+V

m a j E Lioc(R ) , div a th~

negative part V_ of V is 6

less than

1 :

lim sup

~-+o YERm

(see

L2loc (Rm) , VEL2loc (Rm) (2.1 ), -form bounded with relative bound E

J V (x-y) x 2-m dx = 0

Ix I < a

( 2.2)

-

A sufficient condition for (2.2) is that V is a sum of Vi • Vi E LPi(Rm) , Pi > ~ • Let 7J (H) = Co'" (Rm) • Hu = Lu for u (H). Then H is essentially selfadjoint (see [41 for V = 0 , the general case follows easily from this). _ As in Section 1 , we choose ex ~ 0 such that T = H + a is posi ti ve definite and define the spaces ~ 1 • ~ -1 • ';}t1 , ';}to usi ng T. Note that ~-1 and -ae,o are distribution spaces, H': ~1 -+ ~-1 is a differential operator (this follows from the fact that C "'(Rm) is dense in ~1 • i. e. it is a form core of H - see [7]) •• 0 Now assume that q is the operator of multiplication by the function iq , {J:) (q) = C; (Rm) • q is a real valued locally square integrable function such that q = q1 + q2 • Iql (x) I ~ C V+(x) (2.3) for some C ~ 0 • V+ = V + V_ , [6]

).

{ I q2 (x-y) II x 12 - m yERm Ix 0, for all fEH and s (A) _y n

)}, where y

=

n

n

If

_

> 0,

cn- r {l+O(n q)}, r,q

=

-1

min {q, raO +ra)

then

}.

REl1ARK 1.

The estimate of the remainder is close to sharp: for the elliptic op2 erators in L (D) the remainder is of order given in Theorem 1. r THEOREH 2. I f A> 0, \(A) - cn- as n + "', r > 0, IQfl i clAafl, 0 < a,

N(I+Q)

=

{O}, and ra

TI!EOREH 3.

If t -

to}, then sn(L+H) REHARK 2. If d then L+MER (H). b

ffi

~

1, then BERb(H).

~

d then L + M£Rb(H) , H

sn(L) {l+O(n

1 then m

1.

Furthermore if N(L+H) -1

, (t-m)(t-m+d)

Therefore if d

=

-1

}.

1 and m

2 then the equiconvergence of the eigenvector expansion for the operator A and the ~oot vector expansion with brackets for the operator B holds. REHARK 3. The meaning of the equiconvergence is as follows. Let g be an arbitrary element of H, {h.}, j = l, 2, ..• , be the system of eignvec:tors of A which ]

forms an orthonormal basis of H, {h.} be the root system of B. J

a sequence of integers m].,

ffi].

+

00

such that I I

Then there exists

n

L (P .-p~)gl I

j=1

]

]

+

0 as n

+

00.

Here

speCTRAL PROPER'J'JES OJ' S(),\lF NONSFLFADjOINT OPliR.'lTORS

P.(P~) J

J

is the projection onto the subspace F,(H.) defined in §l. J

J

351

That is equi-

convergence means that the eigenvector expansions and the root vector expansions with brackets converge or diverge simultaneously. For the first time the equiconvergence theorem for the Fourier series and for the eigenfunction expansions for a regular selfadjoint Sturm-Liouville operator was proved by A. Haar (1910) and M. Stone (1928). Since then there were many results in this field but they were obtained for selfadjoint differential operators and in most cases are based on some study of the asymptotics of spectral functions of these operators [6]. The above result is of abstract nature and deals with nonselfadjoint operators. APPLICATIONS Consider the following scattering problem: 2 +k ) u = 0 in [I, u = 0 on S, u = u + v,

('i

(1)

k > 0,

o

DC R3 is a compact domain with a smooth boundary S,

where

v satisfies the radiation condition,

[I

U

3 = R 'D.

exp {ik(n,x)},

o

If one looks for the solution of (l) of the form v = Jsg(x,t)f(t)dt, -1

g = exp(iklx-tl)(4TIlx-tl) , then Af = -u ' where Af = Jsg(s,t)fdt, SES. The O 2 operator A on H = L (S) is nonselfadjoint. Its spectral properties are of interest [3]. One can use theorems 1, 2 for studying these properties. Consider the problem 2

2

3

[V+k-q(x)]u=OinR, with the same u

o

u=uO+v,

k>O,

(2)

as above and with v satisfying the radiation condition.

that q(x) is compactly supported, q(x) = 0 if Ixl > R, qEC~. tion for u is u = Uo - Jg(x,y)q(y)u(y)dy

= Uo

J=

- Tu, 2

The integral equa-

JIYI~R'

Here T is a compact operator on II = L (DR)' DR per ties are of interest.

Assume

(3 ) {x:

Ixl

< R}.

-

Its spectral pro-

2

Namely i t is of interest of know i f TERb(H), H = L (DR)'

PROBLEMS 1)

1

2

2

Let Bf = Ll exp {i(x-y) }Edy be an operator on II = L ([-1,1).

known i f

BE~(H).

It is not

2) If d > 1 i t seems to be an open problem i f L + MER(H) under

the assumption of Theorem 2. Is the bracketing necessary? Some other problems can be found in [3) and [5], where some questions of interests in applications are also discussed. SKETCHES OF SOME PROOFS 1)

Theorem 1.

Let U = A*A, V = B*B = (I+Q*)A*A(I+Q), Ln be the linear span of

(I+Q)f and Mn = (I+Q*)L • Then the condition glL n n is equivalent to f 111n' where 1 means the orthogonality in H. From the minimax

n first eigenvectors of U, g

princple it follows that

352

A.j. RAMM

< sup -

(Vf,f)

If1 2

flL

(Ug,g) < sup cg;g)

-

f1M

fllr n

.;~ m

(g,g) I;j2 -

n

2 (A) {I + sup sn+1 flL m (4 )

(I+Q)-1, S

Taking into account that U ; (I+S*)V(I+S), where 1+ S we conclude that 2 2 a sn+1+2m(A) i sn+1+m(B) {I + O(sm(B»}. From (4) and (5) it follows that s (B)s-l(A) n

2 sn+1+2m(A)

s

S~+l+m(A)

m

(A)

n+m Sn(A)

n+m

provided that mn s

(B)

n+1+m sn+l+m(A) = 1

=

-1

+

O.

+

00, and


O.

(7)

Then (6) and (7) imply that

1 + 0 (n-(l-x)ra) + O(n- q ) + O(n-x)

+ O(n- Y),

(8)

where

Y = min {q,(l-x)ra,x}

min {q,ra(l+ra)

-1

}.

(9 )

Theorem 1 is proved. It is known [7], that -1

-1

d2 2 N(A) = cA (l + O( A)} (10) where N(A) is the number of the eigenvalues An of an elliptic selfadjoint opera2 d tor L on II = L (D), D C R , ord L = 2. Thus td- l _d- 1 An ; c n {1 + O(n )}, (11) 1

because An is the inverse function with respect to N(A). remainder in (10), and therefore in (11), is sharp. (L+M)

-1

=

A(I+Q) , Q = -(I+T)

sn(A) {I + O(n a

= (2-m)2-

and 2 - m O(n28].

d

-1

1

-Y

-1

)}, Y = min {d

T, T ; ML -1

2-m '2-m+d}.

, and we used formula (9).

> d(d-1)-1

-1

,A

=

-1

L

The estimate of the

For the operator B; Theorem 1 says:

In this case

Therefore Y

=

I' ;

2d

-1

2-m 2-m+d if d

sn(B)

,q =

1.

If d

>1

then the estimate of the remainder given in Theorem 1 is

-

) and it is sharp.

The first statement of Theorem 1 was proved in [3, p.

353

SPECTRAL PROPERTIES OF SOME NONSELFADjOIN'j' OPliRA'fORS

2) Theorems 2, 4. In [3], Appendix 11 the following proposition was proved: assume that L > 0 is an operator on a Hilbert space H with a discrete spectrum r

l

), r < r, r > 0, and ITfl ~ clLafl, a < 1, where T is a linear l (nonselfadjoint) operator; i f r(l-a) ~ 1 then L + TS~(H), i f r(l-a) ~ 2, then Aj

=

cjr + O(j

the eigenvector expansion for the operator L and the root vector expansIon for the operator L + Tare equiconvergent. then BSRb(H).

Let A-I = L, B-

1

= L + T.

He have B = (L+T)-l = A(I+TL-l)-l = A(I+Q) , Q

The operator (1+TL

-1 -1

)

is bounded.

I f B-ISRb(H)

l _(I+TL- )-l TL -l. a

Therefore the inequality IQf I ~ c I A f I im-

plies that ITL-il < cIL-afl, or ITfl < CILl-afl. Thus a = I - a and the conditions r(l-a) ~ 1, r(l-a) ~ 2 are equivalent to ra ~ 1, ra ~ 2 respectively. 3) Theorem 3. The argument given after formula (10) proves the second statement of Theorem 3. The first statement of this theorem follows from Theorem 2. In1

deed r = £d- , a = (£_m)£-l and the condition 1 < ra can be written as £ - m> d. This condition implies that L + NSRb(I{). 4) Applications to scattering theory. The operator A defined in n.4, can be written as A = Al + iA2 = ReA + iImA, where Al (A+A*)/2, A2 = (A-A*)/(2i). The kernel of Al is cos {kl s-t I} (411 I s-t I) -1, k > 0, while the kernel of A2 is sin (k I s-ti ) (4111 s-t I) -1.

The operator Al is an elliptic pseudo-differential op-

erator of order -1, while A2 has the order -00: -1

ator.

-1

Suppose that Al

IQfl ~ clA~fl with a mate).

and A



(f~O),

... ,f(Zn-l) (0»

Let L[y] = Tj"=D(-l)j (Pjy(j»

0, Pn ~ ~ > D.

and

(j) with each

If

S is an n dimensional subspace of C2n such that

f, gEC~[O,oo), f(O)E S, g(O)E S ~

f~ L[f]~ = f~ ~~=OPjf(j)~(j), and (b)

(2 )

there are exactly n linearly independent Dirichlet so-

lutions of L[y] = 0, then the Friedrichs extension HS of the positive symmetric. operator TS with domain {fECoo[D,oo) ---c

: f(O) ES} satisfies

domain HS

{f

domain Hl/2 S

{f:f(n-l)E AC loc ' (f,f)D < 00, TInf(O)E TInS}.

E

domain Lmax:

(f, f) D

< 00,

f (0) E S},

This result has been extended to a wider class of expressions in a weighted Hilbert space by Bradley, Hinton, and Kauffman [1].

The

characterization of domain H~/2 was obtained earlier by Hinton [11] under the assumption that L have exactly n square integrable solutions.

It is also shown in [1] and [13] that if condition (b)

358

THOMAS T. READ

fails, then the domains of HS and

H~/2

are proper subsets of the

indicated sets. Condition (a) of Theorem 3 is a restriction to a certain class of symmetric boundary conditions.

It can be seen to be unnecessary by

showing that for any n dimensional symmetric boundary space S the difference between the two sides of (2) can be written as a quadratic form in

TI

n

f(O) and

TI

n

g(O) and so can be estimated in terms of

arbitrarily small multiples of (f,f)D and (g,g)D and some mUltiples of the L2 norms of f and g. Thus the results of (1) and (13) actually hold for arbitrary symmetric boundary conditions. Condition (b) in Theorem 3 is certainly satisfied in the context of Theorem 2, for it is precisely the assertion of Theorem 1.

Thus

Theorem 2 can be proved in the same way as Theorem 3, with the only alterations necessary being those required to accommodate arbitrary symmetric boundary conditions.

H~/2

One application of the characterization of the domain of

in

Theorem 2, discussed at length in (1), is to the minimization of the quadratic functional Q(f) over the set

=

1;(lf"1

2

+

P l lf'I

2

+

POlf12)

~ of all functions f with L2 norm one for which the

Dirichlet integral is defined and finite, and the boundary values (f(O), f' (0)) lie in some specified subspace So of 2,

~

H~/2

~2.

By Theorem

is precisely the set of elements of unit norm in the domain of where S is any two dimensional symmetric subspace of

C4 such

TI S = SO. Thus if S satisfies condition (a) of Theorem 3, then 2 the infimum of Q(f) for f E~ is the least point of the spectrum of

that HS·

It follows that the infimum of Q(f) may be determined from the elements of C~[O,oo) satisfying f(O)E S, that is, from the domain of the symmetric operator TS of Theorem 2. boundary condition is f(O)

In particular, when the

= f' (0) = 0, then the infimum of Q(f)

over ~ is equal to ·the infimum of Q(f) over the C'Q[O,oo) functions with unit norm.

It should be emphasized that this equivalence

depends on the result of Theorem 1, for if the equation L[y) = more than two Dirichlet solutions, then

~

°

had

would include functions

not in the domain of H~/2, and the infimum of Q(f) over ~ might be less than the infimum of Q(f) over the elements of domain TS of unit norm.

DlRTCHLET SOLUTI()NS OF [(){ 'RTII ORiJliR DIFFERENTI:lL EQ(,':lTIONS

359

REFERENCES: [1)

Bradley, J.S., Hinton, D.B., and Kauffman, R.M., On the minimization of singular quadratic functionals, Proc. Royal Soc. Edinburgh, to appear.

[2)

Devinatz, A., On limit-2 fourth order differential operators. J. London Math. Soc. (2) 7 (1973) 135-146. Devinatz, A., Positive definite fourth order differential operators, J. London Math. Soc. (2) 6 (1973) 412-416.

[3) [4) [5) [6)

[7) [8]

[9] [10)

[11)

Eastham, M.S.P., On the L2 classification of fourth-order differential equations, ~r. London Math. Soc. (2) 3 (1971) 297-300. Eastham, M.S.P., The limit-2 case of fourth-order differential equations, Quart. J. Math. 22 (1971) 131-134. Evans, W.D., On non-integrable square solutions of a fourth order differential equation and the limit-2 classification, J. London Math. Soc. (2) 7 (1973) 343-354. Everitt, W.N., Some positive definite differential operators. J. London Math. Soc. 43 (1968) 465-473. Everitt, W.N., On the limit-point classification of fourth order differential equations, J. London Math. Soc. 44 (1969) 273-281. Hinton, D.B., Limit-point criteria for differential equations, Canad. J. Math. 24 (1972) 293-305. Hinton, D.B., Limit-point criteria for positive definite fourth order differential operators, Quart. J. Math. 24 (1973) 367-376. Hinton, D.B., On the eigenfunction expansions of singular ordinary differential equations, J. Differential Equations 24 (1977) 282-308.

[12]

Kauffman, R.M., On the limit-n classification of ordinary differential equations with positive coefficients, Proc. London Math. Soc. (3) 35 (1977) 496-526.

[13]

Kauffman, R.M., The number of Dirichlet solutions to a class of linear ordinary differential equations, J. Differential Equations 31 (1979) 117-129. Robinette, J., On the Dirichlet index of singular differential operators, in preparation

[14] [15]

Walker, P.W., Deficiency indices of fourth order singular differential operators, J. Differential Equations 9 (1971) 133-140.

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Spectral Theory of Differential Operators I. W. Knowles and R. T. Lewis (eds.) © North-Holland Publishing Company, 1981

SPECTRAL AND SCATTERING THEORY FOR PROPAGATIVE SYSTEMS Martin Schechter Yeshiva University New York, New York

We discuss a system of equations that describes many (if not most) wave propagation phenomena of classical physics. We consider spectral and scattering theory under minimal assumptions on the coefficients. 1.

INTRODUCTION.

Many wave propagation phenomena of classical physics are governed by systems of partial differential equations of the form (1.1)

E( x )

n

<JU

\'

-;:-It =

,

j

L

=1

A ~ _ -iAu j

dX

j

where x = (xl"" ,x ) E lRn, u(x,t) is a column vector of length m describing the n state of the medium at position x and time t (cf. Wilcox[27]). Here E(x) and the Aj are real, symmetric m x m matrices with the following properties: (a) (b)

E(x) is a uniformly positive definite function of x; the A. are constant. J

From the point of view of spectral and scattering theory it is desirable that solutions of (1.1) be of the form u

=

e- itH u0'

where H is a selfadjoint operator. This would require that H be an extension of E-1A. vihen E = 1, one can easily obtain a selfadjoint realization Ho of A in H = [L2]m using Fourier transforms. On the other hand, if E f 1, the operator E-1A need not be Hermitian on H. However, it is Hermitian on the Hilbert space Hl with scalar product (1 .2)

(u , v) 1 =

f

v (x) * E(x) u (x)

dx .

(The asterisk denotes the conjugate transpose.) When E(x) is uniformly bounded, one can show that the operator E-1H is selfadjoint on Hl (cf. [27]). However, o when E(x) is unbounded, this operator need not be selfadjoint. It is not even clear that E-1A has an extension that is selfadjoint. 361

362

MARTIN SCHECHTER

We shall study the system (1.1) without assuming E(x) bounded and assuming very little more than (a) and (b). This leads to some difficult technical problems. It is surprising that one can obtain any results at all. However, we show that one can get results for such systems comparable to those known for systems obeying more stringent conditions. Our next step is to add the usual stronger hypotheses to A but still allowing E(x) to be unbounded. We are then able to improve the results. The more we assume about A, the stronger the results become. However, at no point are we required to assume E(xl bounded. 2.

THE EXTENSION.

Our first difficulty is on H . It is not even clear l one, we do not know how many with the problem of deciding is given by Theorem 2.1.

due to the fact that E-1H o need not be selfadjoint that it has a sel fadjoint extension. Even if it has it has. If it has more than one, we are confronted which one to choose. Our solution to these problems

There exists an extension H of E-1A determined uniquely by f and A.

If there are constants a, C such that

J

(2.1)

IE (x) Idx

~ C(1 + Ix I )a

I x-y 1 (t)

(cj>( t)

satisries

r

a

11¢(t)1I dt


1 Then: 3 (i) There are exactly 3 eigenval ues Al ~ '\2 ~ A3 in the interval [2K O,2K l J counted with multiplicity) and '\4 ~ 6K O ~ 2K l · (ii) EqualitY'\l = 2KO or'\3 = 2Kl or'\4 = 6K O implies that (M,g) is isometrically diffeomorphic to the standard sphere S2(K). The techniques for the proof which we developed in [2J come from the fact that eigenfunctions and eigenvalues on standard spheres can be characterized by certain systems of partial differential equations (cf. [7J, [lOJ, [5J); the 371

372

UDO SIMON

pinching of the sphere suggests that we define related expressions. In the following we are going to demonstrate that in higher dimensions(n::.3) our method works on a large class of spaces, which in particular contains many homogeneous spaces (cf. [4J, [6J). 1. Definition. Let (M,g) be connected; we call the Ricci tensor of (M,g) cyclic if VkR ij + ViR jk + VjR ki = O. The following lemma is the basic tool for an integral inequality. 2.

Let (M,g) be connected, n ~ 2, f E Coo(M).

Lemma. [8J.

l:iIlIIHess(f)112 = 2

L

Then

Kij(si - Sj)2 + (Hess(f),Hess(M)) + IlvHess(f)1I 2

i <j

f

ij k

f {2V Rjk - VkR ij }, i where sl,···,sn are the eigenvalues of Hess(f), El,···,E n are corresponding orthonormal eigenvectors in the tangent space and Kij = K(Ei,E j ) is the sectional curvature defined by span (E.,E.).,..; ( ,) denotes the inner product on correspond1 J 1TJ ing tensor spaces, induced by the metric of (M,g). +

3. Computations. We use (2) to prove an integral inequality for fixed f E EA' As the nodal set for f E EA is nowhere dense [3], the zeroes of 1 grad(f)1I 2 are nowhere dense and therefore there exists a uniquely determined continuous function R* on M such that Rijf.f. = (n - 1)R*lIgrad(f)11 2 . J

1

Denote the eigenvalues of the Ricci tensor by r l ~ r 2 ~ ... ~ rn and let (n - l)r: = min r l (p), (n - 1 )r*: = max rn(p), so KO = r = R*(p) = r* = Kl for p E M. Define the symmetric (3.0) tensor B(f) by (3.a) B(f)ijk:

=

f ijk +

~(\+ 2R*)gi/k

+

~(\ - nR*)(gikfj + gjkfi);

then (cf. [2,(3.3)J in the special case of an Einstein space) (3.b) fIIB(f)1I2 jllvHess(f)1I 2dw - ~(3A2 - 4(n - 1 )AR* + 2(n - 1 )nR*2)G, where G: = jllgrad(f)1I 2 dw. (3.c) For the following computations we assume KO > 0 on (M,g), which implies that the Ricci tensor is positive definite; but most of the computations are true without this assumption. From [2, Lemma 1 .2J we get for f (3.d)

E\

2 (A - (n -1 )r*)G ~ jIlHess(f)11 dlu ~ (A - (n -1 )r)G.

Analogously [2, (1.3)J implies for f (3.e)

E

2 j

L K.. (S.

i <j

1J

-

1

For a cyclic Ricci tensor we have

E

EA

s.)2 dw J

>

-

2(n - l)KO(A - nr*)G.

liS1111HTnS H)R UCL\'J'\U,'ES OF THn LAPLACUN

373

where we use the Ricci identity to prove the right hand side inequality. Integration of (2) and the computations in (3) imply the following integral inequality. 4. Lemma. Let (M,g) be closed and connected, n ~ 3, with positive sectional curvature and cyclic Ricci tensor. Then for f E EA (4. 1 ) 0 ~ JI B(f) I 2 dw + P(A) G, where PtA) is the following polynomial of second order l-n 2 4(n+3) n-l 2 (4.2) PtA): = n+2 A +(n-l)(2K +5r - (n + 2) r*)A+2n n+2(r - (n+2)K r*) O o + 4 (n - 1 ) 2 r* (r - r*). Under suitable curvature conditions (e.g. pinching conditions) the polynomial PtA) has two real zeroes A(1), A(2) which depend on the geometric data n = dim M and the curvature bounds K ' r, r*. As PtA) > 0 for A E (A (1) ,A (2)), the integral O inequality implies immediately the following result. 5. Theorem. Let (M,g) be closed and connected with positive sectional curvature and n > 3. If the Ricci tensor is cyclic, there is no eigenvalue in the interval (A(l),~(2)). We were interested in demonstrating how to get integral inequalities; we restrict ourselves by giving explicit values for the above interval in the case of Einstein spaces; then A(1) = nR and 1.(2) = 2(n + 2)K - 2R. The interval (A (1) ,A (2)) is nonempty only if 2KO ~ R, or 6 ~ ~ ,respectively. In [2] we proved that (M,g) must be a Riemannian sphere if an eigenvalue A fulfills A = A(l) orA=A(2). Similar computations can be made for conformally flat spaces (n ~ 3); then L: = Ric - y,Rg fulfills Codazzi conditions, so estimates similar to (3.f) can be given. D. Barthel and R. Kumritz [1] have shown that our technique works also when one considers the related situation for the Laplacian plus a potential on Einstein spaces (because of a result of Cheng [3] their assumption on the nodal set of the eigenfunction is superfl uous (cf. [9], §5)).

[1]

REFERENCES .. Barthel, D. and Kumritz, K., Laplacian with a potential. Proceedings Colloquium Global Analysis - Global Differential Geometry. TU Berlin 1979. Lecture Notes Mathematics 838. Springer, 1981.

[2]

Benko, K., Kothe, M., Semmler, K.-D. and Simon, U., Eigenvalues of the Laplacian and curvature. Colloquium Math. 42, 19-31 (1978).

[3]

Cheng, S.-Y., Eigenfunctions and nodal sets. (1976) .

Comm. math. Helv. 51,43-55

374

UDO SIMON

[4J

D'Atri, J. E. and Ziller, W., Naturally reductive metrics and Einstein metrics on compact Lie groups. Memoirs AMS 18, No. 215 (1979).

[5J

Ga1lot, S., Varietes dont le spectre ressemble a celui de la sphere. Comptes Rendus Acad. Sci. Paris 238, 647-650 (1976).

[6J

Gray, A., Einstein-like manifolds which are not Einstein. Dedicata 7, 259-280 (1978).

[7J

Obata, M., Certain conditions for a Riemannian manifold to be isometric with a sphere. J. Math. Soc. Japan 14, 333-340 (1962).

[8J

Simon, U., Isometries with spheres.

[9J

Simon, U. and Wissner, H., Geometry of the Laplace operator. Kuwait Conference on Algebra and Geometry, Feb. 1981. Proceedings. To appear.

Geometriae

Math. Zeitschrift 153, 23-27 (1977).

[lOJ Tanno, S., Some differential equations on Riemannian manifolds, J. Math. Soc. Japan 30, 509-531 (1978).

Spectral Theory of Differential Operators I.W. Knowles and R. T. Lewis leds.) © North·Holland Publishing Company, 1981

THE SQUARE INTEGRABLE SPAN OF LOCALLY SQUARE INTEGRABLE FUNCTIONS Philip W. Walker Department of Mathematics University of Houston Houston, Texas 77084 U.S.A.

A method using limits of inverses of Gram type matrices is given for determining the square integrable span of a finite sequence of locally square integrable functions. Whenever S is a Lebesgue measurable subset of the real numbers L2 (S) will denote the Hilbert space of equivalence classes of complex valued functions y defined on S with the property that (

J

lyl2

s

exists and is finite. We suppose that each of E and Ek for k = 1, 2, able subset of the real numbers, that for k = 1, 2, . . . ,

is a Lebesgue measur-

and that U E

k=l

k

=

E

We take n to be a positive integer and each of Y1' . . . , Yn to be an equivalence class of Lebesgue measurable complex valued functions defined on E. We suppose that Yi is in L2(Ek) for i = 1, . . . , n and each positive integer k and that (Y1, Yn) is linearly independent over E1. hence over each Ek and over E . Let V be the spa n (i. e . the set of linear combinations) of (Y1' and let L be L2 (E) 11 V It is our purpose here to give a method for determining the space L. The principal difficulty lies in the fact that non-trivial linear combinations of function classes not in L2(E) may be in L2(E). For an example consider ([nJ, [Y2J) where Y1 (x) = 1 and Y2(x) = 1 + l/x for all x?: 1 . The problem of finding L occurs in the study of singular differential operators. See, fot example, [lJ and [2J Theorem XIII.3.8. Earlier results concerning the dimension of L may be found in [31 and [4J. (We caution the reader that in [4J "span" is used to denote the dimension of L.) Here we go further in that we give a constructive way of representing a spanning set for L For each Lebesgue measurable subset f = (f1, . . . , fQ) of members of matrix whose (i,j) entry is

S of the real numbers and each n-tuple L?(S) we will denote by G(S,f) the n

I

S

Thus

G(S,f)

~

f.f . . J 1

is the transpose of the Gram Matrix of 375

(f1, . . . , fn) .

x

n

376

PHILIP WALKER

G(S, f)

Obviously

is Hermitian and G(S, f)

(1 )

=J

f*f

S

bn ) t , where t where * denotes conjugate transpose. If b = (bl' notes transpose, is an n x 1 matrix of complex numbers and v = bl f 1 + bnf n then v = fb so v = b*f* and from (1) we have that (2) vv = b*G(S,f)b .

de. +

j[S

Since the left side of (2) is non-negative we see that G(S,f) is non-negative definite. If (fl, . . . , fn) is linearly independent over S and at least one bi is not zero then v F 0 so the left side of (2) is positive. Thus independence of (fl, . . . , fn) implies G(S,f) is positive definite hence non-singular. When this is the case G-l(S,f) will denote (G(S,f))-l. Our main result is given in the following theorem. THEOREM.

There is an

n

x

M such that

n matrix

lim G-1(Ek'(Yl' k-too

Moreover if

(u 1 '

, un)

i2. given Qy

(u 1 ' .

un) PROOF. G( E ,· ) k

un) = (Y 1 , is L

We will denote by G (.) •

Suppose that

by

by

y,

u, and

k j

~

k.

Then Gk(y) - Gj(Y)

G(E k \ Ej , y) ~ 0

=

so (3)

Hence from

Lemma

Thus

[5J

(See

Gj(Y) 0 page 263.) there is an n x n matrix lim Gk1(y) = M .

M such that

k-too

Since each

Gk is Hermitian so is M. From Lemma 2 below we have that if j s), and apply Theorem 2 to g with c > s. If we express the series as a Stieltjes integral, we obtain 2 2 t /w(x)g(x)¢(X,t)dJ do Ct) = /{p(gl)2 + qg2} - f (a)cot a. (3.3) c _00 ~ ) a

X"

t

f

A calculation shows that the integral on the right is equal to 2 s 2 ~J~ + J ~\ (p(fl)2 + qf2) + ~l~ s-a as-a) s-a

S(l _~\f2 s-a) p,

(1 _

!

under the assumption that pEACloc[a,b). By the ReIly-Bray theorem, as c -; b = co T

I

T > max (O,~). we have

Let ~

f\a

J t Js w(x)g(X)¢(X,t)dX)2 do (t)

~

c

-+ /

~

If we make

proaches

s -+ b = F(t) in

00

t

fj

\~

S W

(X)g(X)¢(X,t)dX)2 do(t)

in the last integral, we find that the inner integral apL2 norm over [~,T] and hence that

JT t ~Js w(x)g(x)¢(x,t)dx)2 do(t) ~

+

JT tF 2(t)dO(t) ~

a

as s + 0 0 . If condition (i) of Theorem 1 holds, then we thus obtain the inequality of the theorem from (3.3) by replacing the limit w of integration by T, letting c -+ b, then s + b and then T -+ b. Clearly, one needs the lemma stated above. If, instead, condition (ii) of Theorem 1 holds, we proceed in a similar way, using g(x) = {I - (x - a)(b - s)/(s - a)(b- x)}f(x) , (a" x s). The theorem is proved in cases (iii) and (iv) by first changing independent variable from xE[a,b) to XE[O,B), where X(x) =

x -1 p ,

J a

B =

j

b

P-

1

(possibly 00),

a

and then using the case (i) and (ii) versions of the theorem on [O,B). The effects of this change of variable are described in sufficient detail in [4] to make it clear how to proceed here. The calculations are quite straightforward.

384

STEPHEN D. WRA Y

REFERENCES [1)

[2) [3) [4] [5J [6] [7J [S) (9) (10)

Amos, R. J. and Everitt, W. N., On integral inequalities associated with ordinary regular differential expressions, in: Differential equations and applications (Proc. Third Scheveningen Conf., 1977), North-Holland Math. Studies 31 (North-Holland, Amsterdam, 1975). Amos, R. J. and Everitt, W. N., On integral inequalities and compact embeddings associated with ordinary differential expressions, Arch. Rational Mech. Anal. 71 (1979) 15-40. Beesack, P. R., review of [1), Math. Reviews SOb:340l7. Everitt, W. N. and Halvorsen, S., On the asymptotic form of the TitchmarshWeyl m-coefficient, Applicable Anal. S (197S) 153-169. Everitt, W. N. and Wray, S. D., A singular spectral identity and inequality involving the Dirichlet integral of an ordinary differential expression, submitted for publication (November 19S0). Kato, T., Perturbation theory for linear operators, 2nd edn. (Springer, Berlin, 1976). Naimark, M. A., Linear differential operators, Part II (Ungar, New York City, 1965) . Putnam, C. R., An application of spectral theory to a singular calculus of variations problem, Amer. J. Math. 70 (194S) 7S0-S03. Sears, D. B., Integral transforms and eigenfunction theory, Quart. J. Math. Oxford (2) 5 (1954) 47-5S. Sears, D. B. and Wray, S. D., An inequality of C. R. Putnam involving a Dirichlet functional, Proc. Roy. Soc. Edin. Sect. A 75 (1975/76) 199-207.