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.
|
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+)
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