Regular Boundary Value Problems Associated with Pairs of Ordinary Differential Expressions
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858 Earl A. Coddington Hendrik S. V. de Snoo
Regular Boundary Value Problems Associated with Pairs of Ordinary Differential Expressions
Springer-Verlag Berlin Heidelberg New York 1981
Authors
Earl A. Coddington Mathematics Department, University of California Los Angeles, California 90024/USA Hendrik S. V. de Snoo Mathematisch Instituut, Rijksuniversiteit Groningen Postbus 800, 9700 AV Groningen, The Netherlands
AMS Subject Classifications (1980): 34 B xx, 47 A 70, 49G xx ISBN 3-540-10706-1 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-10706-1 Springer-Verlag New York Heidelberg Berlin This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to "Verwertungsgesellschaft Wort", Munich. © by Springer-Verlag Berlin Heidelberg 1981 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2141/3140-543210
Preface Numerous papers have been devoted to the study of eigenvalue problems associated with pairs operators.
L,M of ordinary differential
They concern the solutions
to boundary conditions.
f
of Lf
=
AMf
sUbject
In an earlier paper [9] we showed how
these problems have a natural setting within the framework of spaces in the direct sum of Hilbert spaces.
In these notes we work
out in detail the regular case, where the coefficients of the operators
Land
M are nice on a closed bounded interval
and
M
is assumed to be positive definite, in the sense that c2 (f,f)2' f
(Mf,f)2
E "" -
, for some constant
c > O.
It is
hoped that this detailed knowledge of the regular case will lead to a greater understanding of the more involved singular case, where Land
M are defined on an arpitrary, possibly unbounded, open
interval. The work of E. A. Coddington was supported in part by the National Science Foundation, and the work of H.S.V. de Snoo was supported by the Netherlands Organization for the Advancement of Pure Research (ZWO). Earl A. Coddington Los Angeles, California Hendrik S. V. de Snoo Groningen, The Netherlands November 1980
Contents Page 1
1.
Introduction
2.
Selfadjoint extensions of
3.
Forms generated by selfadjoint extensions of
MQ
36
4.
Hilbert spaces generated by positive selfadjoint extensions of MQ'
54
5.
Minimal and maximal sUbspaces for the pair
64
6.
Intermediate sUbspaces
70
7.
Spectra and eigenvalues
106
8.
Resolvents
138
9·
EigenfUnction expansions for selfadjoint sUbspaces
168
10.
Semibounded intermediate sUbspaces
183
aa.
Some special cases
205
References
220
Index
MQ
2l
L, M
224
Introduction.
1.
It is well known that two hermitian tive definite,
H> 0,
to consider
where
matrices
(f,g) = g* Hf,
Then the operator
can also be investigated in
A
Where
f,g
n C,
Of course su
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