Printed in Great Britain PII S0898-1221(98)00210-7 0898-1221(3)

I'Im(Jk'(Xm'{-1) Umholds This implies

:~-Im'{'l()~,(xm'{'2\ Ura..}-I . . . . .

/~/m+2n-l(*~) ( Um-{-2n-iXm+2n) - .~ 0

Ym+l --'--Ym+2 ----""" ---- Ym+2n ---- O,and since y is a solution of a linear difference equation of order 2n being zero at 2n consecutive values, it has to be zero always, i.e., ya=0,

for aU l - n< k< N+ l,

and x= 0 on J* follows. Controllability of (HA) on J* now implies x= u= 0 on J*. Thus (Ha) is strictly controllable on J* with strict controllability index (no smaller index works)

~s=2n-lE

J.

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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

Hamiltonian Eigenvalue Problems

185

4. C H A R A C T E R I Z A T I O N OF EIGENVALUESTHEOREM 2. (Characterization of Eigenval

ues.) Let A 6 R, and let (X, U) and (X:, 0) be normalized conjoined bases of (H~). Then, A is an eigenvalue of (E) if and only if the 2n x 2n-

matrixA:= R*

( xo~o)+~(~o ON+I Oo)XN+I )(N+I UN+I

is singular, and then def A is the multiplicity of the eigenvalue A.PROOF. Let (x, u) be a nontrivial solution of (HA). We put

~:=

(xo~o~(~o)= (~:~~o~] (~o)~o Vo Oo/~o -Vo x: uo

(observe that Definition 1 yields the invertibility of the occurring matrix), and thus

(,,) (x,~,)uk= Uk Ok

d,

fork J*,

since the initial value problem under consideration has a unique solution (observe that I - Ak(A) are assumed to be invertible matrices for all k E J). Now, we have

={R* (x-NX+° 1

~,,+,)+R(21 ON+l)} Oo --x~00)

d= Ad.

Thus, (x, u) solves (R), i.e., A is an eigenvalue of (E), if and only if Ad= 0 holds with d# 0, and this proves our assertion.| Next we wish to simplify this criterion in the case of so-called separated boundary conditions. By this we mean that the boundary conditions (R) may be equivalently written with 2n x 2nmatrices

~.(~

0)

,,a~-(~I T

0

R~/+I

0

RN+I

'

such that the n x n-matrices Ro, R~, RN+I, R~V+l satisfy (as usual) rank(R0

P~)=rank(RN+l

R~v+l)=n,$

P~l~"= P~P~,

T RN+IRN+I= RN+IRN+I.

In this special case, we have the following result.COROLLARY i.

(Separated Boundary Conditions.) Assume that separated (and seg-conjoined)

boundary conditions are given. Let (X, U) be the conjoined basis of (HA), A 6 R, with

Xo= - P~

and

Uo= P f .

Then, A 6 R/ s an eigenva/ue of (E) if and only if the n x n-matr/xR~+IXN+I+ RN+IUN+I

s~.

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

186

M. BOHNER

PROOF. Let A G R. For the above conjoined basis of (HA), there exists another conjoined basis (~:,0) of (HA) so that (X, U) and (X, 0) are normalized (seeL e m m a 1). According to Theorem 2,A is an eigenvalue of (E) iff

~--"" (x-2~N+I)+.(~N~:I ON+i)~0 0o __(_o RN+ 1 )(X~il _~0~ o o 0 RN+I= ( -~

+~

k UN+I UN+I )

v:~o - x: Oo --I

\ R~+IXN+I+ RN+IUN+I R*~+IRN+I+ RN+ION+I]

(=

o

RN+IXN+I+ RN+IUN+I

RN+lRN+,+ RN+,ON+,]

h

(observe Definition 1) is singular, and this happens iff R~+xXN+I+ RN+xUN+I is singular. We wish to conclude this section with the following exanlple. EXAMPLE 1. Let n= 2 and consider the eigenvMue problem (E) given by

A=

(~ 1)0

'

~.~+1~(lo 0)1,

.=(0° 0)~(:~).~(~' o,~) .~+~--(~°o)1 '

According to Remark 3, (V2) is satisfied and (Vx) holds provided N _> 2n - 1= 3. However, now we let N= 2. Then, due to Corollary 1, A E R is an eigenvalue of (E) iff X3(A)=

4A-6 3-2A)4A 6 3 2ARESULTS

is singular. Therefore R is the set of eigenvalues of (E).5. T W O AUXILIARY

In this short section, we cite two results that will be needed in the proof of Theorem 1. LEMMA 2. (Index Theorem; see[19, Theorem 3.4.1].) Let ra G N, let there be given m x mmatrices R, R*, X, U with rank(R R*)=rank(X T uT)=rrt and RR*r= R*R T,

xTu=uTx,

and let X(A), U(A) be m x m-matr/x-va/ued functions on R with XT(A)U(A)= U'

r(A)X()~),x (~ ) -~ x,

A G[Ao -¢, A o+¢],u(~) -- u, as~ -~6, A0+

[or some 6> 0,

,~o,

X(A) invertible for a//A E[A0 Suppose that

6]\{Ao}.

U(A)X-I(A) decreases strictly on[Ao - 6, Ao) and on (Ao, Ao+ e], and denote for A 6[Ao - e, Ao+¢]\{Ao}M(A)=

R*R"r+ RU(A)X-I(A)R T,h= RX+ R'U.

A(~)= RX(~)+ R'U(~),

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

Hamiltonian Eigenvalue Problems

187

Then, ind M ( A o )=

lirnA_.x{ind M(A)} and indM(Ao+)= limA_A+{indM(A)} both exist,°

A(A) is invertiblefor al] A e[A0 - 6,A+ 6]\{A0} for some 6 e (0,~),and the formula def h= ind M ( Ao - ind M (Ao )+ def X+)

holds.

PROOF. W e referto[19,Theorem 3.4.1] (observe also[19,Corollary 3.4.4]).

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LEMMA 3. (Reid Roundabout Theorem; see[14, Theorem 3].) Suppose the system (H) is contro//able on J* (see Definition 3). Let (X, U) and (X, U) be the special normalized conjoined

bases of (H) at 0 (see Definition 1). Then,~> 0 (see Definition 2) if and only if

KerX~+t c KerXk,I

XkX~+l(I - Ak)-IBk>_O,

for all/c E J,

M:=R*R T+ R

UN+I

UN+I

0)(0

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