Spectral Theory and Asymptotics of Differential Equations, proceedings of the Scheveningen conference on differential equations

203 Pages·1974·5.163 MB·iii-vi, 1-210\203

by  E.M. De Jager (Eds.)

NORTH-HOLLAND
MATHEMATICS STUDIES 13
Spectral Theory and
Asymptotics of Differential
Equations
proceedings of the Scheveningen
conference on differential equations,
The Netherlands, September 3-7,1973
E. M. DE JAGER
University of Amsterdam
1974
NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM LONDON
AMERICAN ELSEVIER PUBLISHING COMPANY, INC. - NEW YORK
@ North-Holland Publishing Company - 1974
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system,
or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording
or otherwise, without the prior permission of the copyright owner.
Library of Congress Catalog Card Number: 14 78465
North-Holland ISBN: S 0 1204 2100 2
North-Holland ISBN: 0 7204 21 13 4
American Elsevier ISBN: 0 444 1064 I 3
PUBLISHERS:
NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM
NORTH-HOLLAND PUBLISHING COMPANY, LTD. - LONDON
SOLE DISTRIBUTORS FOR THE U.S.A. AND CANADA:
AMERICAN ELSEVIER PUBLISHING COMPANY, INC.
52 VANDERBILT AVENUE, NEW Y,ORK, N.Y. 10017
PRINTED IN THE NETHERLANDS
PREFACE
These proceedings form a record of the lectures delivered at the Conference
on Spectral Theory and Asymptotics of Differential Equations held in Sche-
vmingen (the Netherlands) from 3 to 7 September, 1973.
The conference was attended by 40 mathematicians from France, Germany,
Sweden, the United Kingdom and the Netherlands.
The number of participants has been limited to 40 in order to give full
opportunity for discussions and exchange of ideas. A list of participants
with their addresses is to be found on page 207 of these proceedings.
All lectures were given on invitation and the table of contents on page vii
gives the titles and the respective speakers.
The Organizing Committee consisted of B.L.J. Braaksma, W. Eckhaus, E.M. de
Jager and H. Le mei. The committee thanks all participants and in particu-
lar the speakers who made this conference so successful.
The committee is very much indebted to the Minister of Education and
Sciences for giving a generous financial support without which this conf-
erence could not be held.
The committee thanks also Mr. M.H.J. Westerhoff and Mr. L.E. Leeflang of
the Department of Education and Sciences of the Government for their help-
ful cooperation in financial affairs.
In these proceedings the texts of the lectures have been put in a certain
order such that lectures dealing with basically related subjects.are
brought together (e.g. 1-9 and 15-19).
Since the famous papers by Hermann Weyl in the "Mathematische Annalen" and
in the "Nachrichten der kgl. Gesellschaft der Wiss. zu G6ttingen" in 1910
on differential equations with singularities, in particular
M cyl = -(py')' + qy = Ary, 0 5 x < -,
there have been published many investigations on this subject, a.0. by
Stone,Titchmarsh, Kodaira, Coddington and Nabark. This topic is still a
subject of lively interest. The contributions 1-9 give recent developments
in this beautiful theory. W.N. Everitt investigates the case where the
weight function r occurring in the differential equation may be unbounded
or oscillatory; H.D. Niessen and A. Schneider consider so called left-
definite systems of differential equations, a generalization of the case
Ir(x)l s pq(x) on CO,-).
J.B. MacLeod investigates conditions in terms of p and q for M to be
limitpoint, M.S.P. Eastham gives examples of second and fourth order
differential equations with oscillatory coefficient q(x) such that M
or its generalization is not of limit-point type and B.D. Sleeman considers
8.0. a generalization of the limit-point limit-circle theory to the multi-
parameter case on which there is also a contribution by F.M. Arscott.
PREFACE
R. Martini deals with differential expressions of the type aD2 + 8D with
a positive on a bounded open interval I, but zero at the boundary of I.
F'leijel's contribution bears upon a positive symmetric ordinary differential
operator combined with one of lower order and it is a generalization of his
work on limit-point and limit-circle theory.
We have digressed here a little bit on the contributions of these authors
because we believe that these contributions together give a kind of "the
state of the art" of current research in the Weyl theory on singular
differential operators and they may serve as an up to date introduction in
this field of mathematical research.
Partial differential operators are dealt with in the papers 10-1~; the
subjects are degenerate elliptic operators in unbounded domains, scattering
theory for wave operators end self adjoint extensions of a Schradinger
operator. Then follow two papers on distribution theory in connection with
differential equations (13-141, one on the connection between certain
spaces of generalized functions and associated linear operators and the
other on quasi analytic solutions of a class of convolution equations.
Finally the lectures 15-19 are again a series of lectures on subjects
which have a common theme viz. asymptotic approximations of solutions of
differential equations. In particular we mention here the paper by
W. Fckhaus, who gives an improvement of the well-known Krilov-Bogolioubov
Mitropolski method for abtaining an asymptotic approximation for non
linear oscillations. The other papers deal with matching principles,
singular perturbations for linear and non linear elliptic equations, and
weakly non linear oscillations.
Because these proceedings exhibit in coherent sequences of papers modern
developments of spectral theory and asymptotics of differential equations,
we hope and expect that this book may stimulate research mathematicians
and advanced students working in differential equations.
The editor expresses his thanks to Mrs. M. van der Werve, secretary in the
department of mathematics of the University of Amsterdam, for her asaiet-
ance as well during the period of organizing the conference as during the
final stage of the preparation of these proceedings.
Amsterdam, February, 1974
E.M. de Jager, Editor.
1
A positive s.ymmetric ordinary differential
operator combined with one of lower order
by
0
Ake Pleijel
[a
The lecture canpletes and refines a communication by K. Emanuelsscn,
Department of Mathematics, Uppsala. Special cases of the problem were
[d, [g [u
treated in and in a so far unpublished joint paper by
C. Bennewitz and the author. The method is a generalization of H. Weyl’s
[g.
famous paper For similar purposes a generalization of the aame kind
[a, [u.
was used in The written version of the lecture was worked
out during a visit to the University of Dundee. For the opportunity to
make this visit the author wishes to thank the Science Research Council,
United Kingdom.
Two formally symmetric ordinary differential operators S-and T are
considered of which S has a higher order than T and a p-ositive Dirichlet
integral. The operators are given on an arbitrary interval I by sums
where D = id/dx and the complex valued functions ajk, b are Sufficiently
jk
regular on I and enjoy hermitean symmetry. It is assumed that
That the definite order M = 2m of S is greater than the maximal order
N of T, in particular means that n < m. By partial integrations over
J = [a,p] CI a fonnula
a
P
is obtained by which the Dirichlet integral (u,v),, or
a
0
Am PLEIJEL
2
is introduced. The Dirichlet integral is a hermitean form determined by
representation (1) of S. The dots in .(2) indicate out-integrated parts.
For the theory it is necessary to consider the relation Su = ?!ti or
more precisely a linear space
M K
E(1) = {U = (u,;) E C (I) x C (I) : SU = 'hi]
of ordered pairs U = (u,;). The auffYcient regularity conditions chosen
in this definition mean that u and have continuous derivatives on I
of all orders 6 M and of all orders 6 K = max(N,m) respectively. Then
a Green's fOn?3Uh
holds for every campact interval J of I when u = (u,6), V = (I?,$)b elong
to E(I). In (3)
while q is an out-integrated part containing derivatives of u,ti,v,i.
By computing q and completing squares it followa that
X
1 1
so that the signature of the hermitean form q i.e. the pair of ita
X'
positive and negative inertia indices, satisfies the inequality
sig q, g (wd.
A consequence of (3j is that
sig Q G (M,M). (5)
J
The solution space Eh(I), where X is a oomplex (in general non-real)
number, is defined by
POSITIVE DIFFEIlENTIAL OPERATOR
3
Ek(I) = )(u,b) EE(I)I.
The elements of this space correspond to solutions of Su = hTu, and
E (I) has the dimension M. On Ex(I) and E-(I) the form (4) reduces to
x h
where in both cases
c = i-'(k- T).
A positive character of the Dirichlet integral is essential for the
theory. The definition of this positivity is related to a class of
compact intervals J = [a, p] of which I can be obtained as a monotonic
limit. The class only contains sufficiently extended sub-intervals J of I
(containing a certain sub-interval Jo). In the sequel the letter J
is reserved for intervals in such a class. Dirichlet integrals can be
formed with functions having m continuous derivatives on I. It is
assumed that for such u the Dirichlet integral
(U,U)~i s non-negative and non-decreasing (8)
J
when J increases, and that
when u is not identically 0 and X is non-real (the vanishing on a sub-
interval of a regular solution of Su = implies ita vanishing on I).
Because of (9) the reduction formulae (6), (7) show that CQ is
3
positive definite on E (I) ard negative definite on E,(I) provided X
x X
is non-real. In particular Q is non-degenerate on these spaces. The
J
Q-projection U(J) on Ex(I) of an element U in E(1) is determined by
J
AKE PLEIJEL
4
The linear subspace E[I] of E(1) is defined by the conditions for
u = (u,li) that
(u,u)s < +OD, < +OD (11)
I I
i.e. that the Dirichlet integrals remain finite when extended over I.
From the definition (4) it follows for J tending to I that the form
exists when U = (u,;), V = (v,#) belong to E[I]. On the similarly
restricted solution spaces
EkII = I ( u , ~ )E E[IIJ,
y 1 = I(.,3 EE[IlI,
reduction formulae of the same type as (6), (7) are valid with the same
i-*(x-X),
c = namely
It follows that cQ is positive definite on Eb[I] and negative definite
I
on L[I]. It shall be proved that CQ is positive definite on a
A I
linear hull iU, Eh[I]] in which U in E[I] does not belong to E h[ I]. In
this way EAII] is maximal, i.e. non-extendable with the indicated property.
This statement generalizes a similar one concerning cQ on EA(I). While
J
the statement for CQ is an easy consequence of (5) the assertion about
J
cQ requires a more elaborate proof. A fFrst step is the deduction of
I
the following
THEOREM. To every U = (u,6) li_n E(1) for whiah
-
(a hu,b-?alJ, <+OD
I
POSITIVE DIFFEXENTIAL OPJTRATOR
5
with a non-real A, there exists a unique U(1) 9 EA(I) mch that
-
U U(1) E E[I], (16)
-
Q(U U(I), E,,[II)= 0. (17)
I
The inequality
cQ(U- U(I), U- U(1)) c 0 (78)
- I
fi c = i-'(A- h) holds true for every U and its related element U(1).
For an indication of the proof, write U = (u,h + f) so that
-
f = and (f,f)S < +m. If J' 2 J a simple computation on account
I
of (4) leads to the formula
-
cQl(U,U)- cQ( U,U) = o2 (u, u ) ~ic ( f,u)S+ ic (u,f)S. (19)
J J J'-J J'-J JIJ
Because of (6) the Dirichlet integral over J'-J is non-negative.
Thus Cauchy-Schwarz' inequality can be used for the last two terms in
(19). This gives the inequality
where the meaning of 11 11 is obvious. In (20) the element U can be
J'-J
replaced by U-V with V = (v,hv) € Ex(I). The difference u-v has the
same form as U = (u, hu + f) with the same f. In this way (20) implies
that
CQ (U-V, IJ-V) + (f,f)S is non-decreasing (21 1
J J
when J increases and V belongs to E (I). The inequality
h
EJ : CQ(U- V, U-V) + (f,f)S d (f,f)SJ V EEx(I), (22)
J J I
defined one side of a second order surface in the finite dimensional
apace E x( I)o r is never satisfied. The sum of the second order terms in
(22) is the expression
0
AKX PLEIJEL
6
cQ(v,V) = C*(V,V)~
J J
which is positive definite according to (9) so that (22) is(the
interior CJ) of an ellipsoid or the empty set. The centre of (22)
is determined by an equality which coincides with the formula in (10),if
in this formula U(J) is replaced by V. This tells that the centre of
(22) is V = U(J). Now the following inequality holds true, viz.
c-Q(U- U(J), U-U(J)) Q 0.
J
For the assumption to the contrary implies that cQ would be positive
J
definite on the linear hull
-
IU,E,,(I)] = jt(U U(J)) + V: t = number, V EE,,(I)].
If U is not in E (I) this violates (5). If U belongs to Eh(I), the
A
inequality (23) is trivial since U(J) = U. The inequality (23) shows
that zJ is non-empty. From (21) it follows that the ellipsoids have
the shrinking property that CJ 2 ZJi when J C J'. The element
U(1) is obtained as the limit of the centre U(J) when J I. This limit
+
is contained in all ellipsoids and the inequality in (22) is satisfied
-
for all Jwhen V = U(1). Thus (18) is valid. It follows that U U(1)
is in E[I]. The relation (17) expresses the fact that U(1) is the centre
- -
of the limit ellipsoid cQ(U V, U V) Q 0, V € EX(I).
I
The Theorem is now applied when U = (u,;) belongs to E[I]. Its
condition (15) is then automatically Wfilled. The element U(1) must
belong to E [I] and is the Q-projection of U on this spice. The
h
1
inequalify (18) shows that Eh[I] is maximal in E[I] with the property
that cQ is positive definite on this subspace. For if U is outside
I
+
Eh[I] the difference U- U(1) is 0 and gives a non-positive value to
CQ (U-U(I), U-U(1)). Thus cQisn ot positive definite on {U, Eh[I]].
I I
In the same way it is seen that E [I] is ta8Xhd as a subspace
5
of E[I] on which CQ is negative definite. A consequence of these
I