Printed in Great Britain PII S0898-1221(98)00210-7 0898-122198 19.00 + 0.00 Discrete Linea(3)

implies z= u= 0 on J*. The minimal integer tcs E J with this property is then called the strict controllability index of (Ha). I

For stating our main results, we wish to label the following assumptions. (V1) (Ha) is strictly controllable on J*. (V2)~1 _<~2 always implies Ht`(~l) _< Ht`(~2) for all k E J. (Vs) There exists~ E R such that (.;~)> 0 and such that )~ _<~ always implies for all k E J Ker Bk(~) C Ker Bk(~) andB~()~){B~(~)-B~(~)}B~(~)>_O.

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

183

Now our main result reads as follows.THEOREM 1. Assume (V~), (V~), and (Vs). Then, ff there exist eigenvalues of (E), they may be arranged by-oo<~AI_~As_~ A3_~""

More precisely, (i) (V1) and (V2) imply that the eigenvalues are/solated, and (ii) (V2) and (Vs) imply that the eigenvalues are bounded below by A (which is not an eigenvalue) provided (HA) is controllable on J* for all A E R.|The remaining sections are devoted to the proof of the above theorem. However, here we wish to make some remarks concerning the concept of strict controllability. REMARK 1. Suppose Ak(A) -: Ae and Bk(A) -: Bk are constant for all k E J. Then condition (ii) of Definition 4 (with strict controllability index s, E J) already implies condition (i), i.e., controllability of (HA) on J* for all A E R, and the controllability indices s(A) of (HA) satisfy maxAeR~(A) _< ss+ 1 E J*. To prove this, assume (ii), let there be given A E R, a solution (z, u) of (HA), and m E J with m+~,+ 1 E J* such thatXn= Xm+l= .Tm+2=

=

Xm+~,+l

--~ 0

holds. Therefore,0 (A)xn+l= 0+l(A)xn+ .....= 0,

andhence(note/-/k(A)=(-C0(A)

0)0 f o r k E J ),

Condition (ii)thus implies x= u= 0 on J* so that controllability of (HA) on J* with controllability

index~(A)<~a+ 1 follows.| R E M A R K 2. Suppose as in the previous remark that Ak(A) and Bk(A) are independent of A E R for all k E J. Furthermore, assume that 0k(A) is nonsingular for all k E J and all A E R. Then controllability of (HA) on J* for all A E R with controllability indices s(A) E J implies strict controllability of (HR) on J" with strict controllability index s,< s:= maxAeR s(A). To show this, let A E R, let (x, u) be a solution of (HA), let m E J with m+ s E J (this yields m+ i E J since s(A) _> 1 trivially),and assume

Then, we haveCn(A)Xn-i-1= Cn-i-l(A)Xm..b2 ..... Orn+~CA)Zrn..[-~q-1= O,

and hence,invertibility of Cn(A),Cm+I(A),..., O,n+~(A)yieldsXm+l -.Tm+2= ' "~--- X r a+~+ l= 0.

Now controllability of (HA) on J* with controllability index~(A) _<~ together with m+ 1 E J and m+ 1+~ E J* imply z= u= 0 on J ' . Altogether, strict controllability of (HR) on J" with strict controllability index~,<_~ follows.|

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

184

M. BOHNER

R E M A R K 3. The purpose of this central remark is to show that Sturm-Liouville differenceequations of order 2n depending linearly on an eigenvalue parameter A R satisfy (V1) provided N> 2n - 1 holds. To start with, let r (~)ER,

We consider the n× n-matrices 0

(01/0<u<n, 1"-."'. 0, .

and

r(k )~ O, n

for a l l k E Z .

(0)... 0 1, k J,

Ak(A)= Ak=

Bk(A)~ Bk=

,r(~")

c~(A)=

_ A

0"..

A R

r (n-l)

0

and the corresponding systems (HA),£ E R. Note that the 2n× 2n-matrices/:/~(A) are given by

/x

)

ilk(A)=

o

,

k J,

0 (so that (V2) is satisfied). Let N _> 2n - 1. Clearly (HA) is controllable on J* for each A R (see, e.g.,[12, Remark 20)]) with controllability index n _( N+ 1, i.e., n J*. Let A R and pick a solution (x,u) of (HA). It is very well known (see, e.g.,[4, Section 3]) that in this case Ayk_l xk=| A2Yk-2, for a l l k J *,

\ An-lyk+l-n holds with a solution yk (1 - n _< k _< N+ 1) of the linear self-~ljoint difference equation of order 2n n

n, ta--0

(SLA)

a so-called Sturm-Liouville difference equation. (In fact, (SLA) and (HA) are even equivalent in the sense that a solution of (SLA) yields a solution of (HA) and the other way around; see, e.g.,[4, Section 3].) Now we assume that for some m E J with m+ 2n - 1 E J

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,

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