Printed in Great Britain PII S0898-1221(98)00210-7 0898-122198 19.00 + 0.00 Discrete Linea

bohner~mr, edu Abstract--This paper introduces general discrete linear Harniltonian eigenvalue problems and characterizes the eigenvalues. Assumptions are given, among them the new notion of strict control-lability of a discrete system, that imply isolated

Pergamon

Computers Math. Applic. Vol. 36, No. 10-12, pp. 179-192, 1998© 1998 Elsevier Science Ltd. All rights reserved Printed in Great Britain P I I: S0898-1221(98)00210-7 0898-1221/98$19.00+ 0.00

D i s c r e t e Linear H a m i l t o n i a n Eigenvalue P r o b l e m sM. BOHNERDepartment of Mathematics and Statistics University of Missouri-Rolla Rolla, M O 65409-0020, U.S.A. bohner~mr, eduA b s t r a c t - - T h i s paper introduces general discrete linear Harniltonian eigenvalue problems and characterizes the eigenvalues. Assumptions are given, among them the new notion of strict controllability of a discrete system, that imply isolatedness and lower boundedness of the eigenvalues. Due to the quite general assumptions, discrete Sturrn-Liouville eigenvalue problems of higher order are included in the presented theory. (g) 1998 Elsevier Science Ltd. All rights reserved.

K e y w o r d s - - L i n e a r Hamiltonian difference system, Discrete Sturm-Liouville eigenvalue problem, Strict controllability, Reid roundabout theorem, Comparison theorem.

I. I N T R O D U C T I O N In this paper, we introduce discrete linear Hamiltonian eigenvalue problems, i.e., eigenvalue problems which consist of a linear Hamiltonian difference system depending on an eigenvalue parameter A E R subject to self-adjointboundary conditions. The m a i n result on these problems states that, under certain assumptions, the eigenvalues may be arranged as follows:--c~< A1<~ A2<: A3<= ' ' ',

i.e., that the set of eigenvalues is bounded below and that the eigenvalues are isolated in the sense that for any A E R one may pick an 6= e(A)> 0 such that the interval (A -~, A+~) contains at most one eigenvalue. The central notion connected to this isolatedness is the new concept of strict controllability of discrete systems which is also introduced in this paper. The m a i n tools on handling these eigenvalue problems and on proving the above result is a theorem that gives a useful characterization of the eigenvalues (in terms of some matrix being singular), an index theorem (which calculates the local change of the number of some matrix-valued function's negative eigenvalues), a Reid roundabout theorem (that characterizes so-called positive definiteness of discrete quadratic functionals), and a comparison theorem (which states that positive definiteness of one functional together with certain assumptions imply positive definiteness of some other functional). Finally, it should be emphasized that our general assumptions allow us to include discrete Sturm-Liouville elgenvalue problems of higher order so that these important problems may be treated with the same techniques. Let us shortly give an overview on the existing literature of the subject. Discrete SturmLiouville difference equations of order two as well as linear Hamiltonian difference systems have been an object of recent interest. Linear Hamiltonian difference systems were introduced by Erbe and Yan in[1] a

nd examined in three proceeding papers[2-4] by the same authors, however, under assumptions that only include the case of Sturm-Liouville difference equations of order two but

Typeset by .A.,~S-'~X.179

bohner~mr, edu Abstract--This paper introduces general discrete linear Harniltonian eigenvalue problems and characterizes the eigenvalues. Assumptions are given, among them the new notion of strict control-lability of a discrete system, that imply isolated

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M. BOHNER

not of higher order. Further important results in this matter have been obtained by Ahlbrandt, Do~l~, Heifetz, Hooker, Patula, Peil, Peterson, and Ridenhour in[5-11]. In a recent series of publications by the author[12-17] (one of them is a joint work with Do~il~), linear Hamiltonian difference systems were considered under assumptions that include the important case of SturmLiouville difference equations of higher order and that give so-called Reid roundabout theorems for those problems. This work may be considered as one origin of the results proved in the present paper. The other origin is the treatment of continuous linear Hamiltonian eigenvalue problems as is done in the paper[18] by Baur and Kratz and in the monograph[19] on the subject by Kratz. This work also contains the above cited index theorem which may be successfully applied in our discrete case also. Finally, while the study of eigenvalue problems in the existing literature basically reduces to discrete Sturm-Liouville eigenvalue problems of order two (see the books by Agarwal[20, Chapter 11] and by Kelley and Peterson[21, Chapter 7]), a special Sturm-Liouville difference equation of higher order depending on an eigenvalue parameter has been considered in the recent paper[22] by Kratz; however, there is no theory for eigenvalue problems subject to general linear Hamiltonian difference systems. A brief discussion of this paper's s e t u p is in order. The following section introduces discrete linear Hamiltonian eigenvalue problems and gives some preliminaries on linear Hamiltonian difference systems. In Section 3, we present the main result of this paper and give the assumptions that are needed; among them we introduce the concept of strict controllability of discrete systems and the notion of the so-called strict controllability index, which has no obvious analogue in the"continuous theory". Section 4 contains a characterization of the eigenvalues, and this characterization is also improved in some sense if the boundary conditions under consideration are separated. While Section 5 recalls two important auxiliary results (the index theorem from[19, Theorem 3.4.1] and the Reid roundabout theorem from[14, Theorem 3]), Section 6 contains a series of lemmas that are needed for proving the isolatedness of the eigenvalues. Finally, a comparison theorem is proved in Section 7, and as an application of it we show that the eigenvalues are bounded below.

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