holds.
PROOF. W e referto[19,Theorem 3.4.1] (observe also[19,Corollary 3.4.4]).
|
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
X N+l invertible,XN+I
XN+I
>0,
onImR
holds.
PROOF. W e referto[14,Theorem 3] and remark that R{RtR*RtR} R T= R * R T holds.
|
6. I S O L A T E D N E S S OF E I G E N V A L U E SIn this section,we wish to establishTheorem 1(i).Consider the followingcondition. For all Ao E R there exists e> 0 such that XN+I(A) is invertible and~N+,(A) (UNI(A) 0 ) (xNOI(A) I ) -1 isstrictlydecreasingfor A E[Ao-e, ON+,(A) Ao+~]\{Ao}, where (X(A), U(A)) and (X(A), 0(A)) are the special normalized conjoined bases of (H) at 0 for every A E R.Of course, condition (I) implies by the index theorem, L e m m a 2 (observe also L e m m a 1 and the
(I)
continuity of the Hk(A), k E J), that the singular points of
A(A)= R, IxNOI(A)=
.~N+I(A)I.jcRIuN+II(A) 0+n
I
R"
(-Xo(~) k~~~ x+()
x+(~°)ku~+~(~)(uo(~)t~JO°(~) N~r(~~/ l 2~ )
~-,+,.,,
i.e.,(according to Theorem 2) the eigenvaluesof (E), are isolated.Therefore our goal is to show that (V1) and (V2) imply (1).This we will achieve by showing some lemmas. L E M M A 4. Suppose (X(A), U(A)) is a conjoined basis of (HA) for eac~ A E R with Xo(A)=
0o(~)= 0. Then,k-1/x~+1(~)~T~(~) (x~+~(~)) X~(~)0k(~)-u~C~):~k(~)= - E\ U~(~) )\ U~(~)m=0
holds for al/k E J*\{0} and for al/A E R.
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
188
M. BOHNER
PROOF. Let A, p R and m J. Then,
n[X~(.){U.(A) - U~(.)} - U~(.){X~(A) - X~(.)}]= n[x~(~)u~(A) - G~(~)x.(A)]= A[ t u.(~) -x.(A)
t u.(~) ) j t t-x-+~(A) )
+;/x.+,(~)
{~(u.cx)
~
={~(-U~(")'TI (x'+r(A))-(X'+~(")'{ T t x.(.); t u.(A) t, u.(.) ) t x.(A) ) )u.(,) )j\ u.(A) ) - t u.(v): ( x,.+,(.) ) T (x.+,(A)~= - I t U.(#){Hm(A)-H,.(#)} U.(A) )" Now, division by A -# and letting/~tend to A yields
s,_x.+,<>,)+ k u~(v) )
( -x.(A)/j (tH'(X)x )}~ X'+'(A)u.(A)
A
~wl
-"w6
j
k u.(A) )
k u.CA) )$
so that )Co(A)= Co(A)= 0 prove the validityof our assertion.
L E M M A 5. Suppose (X(A), U(A)) and ()[(A),0(A)) are norma!i~,ed conjdned
bases of (HA) for
A e R with Xo(A) -- 0o(A)=~o(A)= bo(A)= o. Let~ a'. A~meinvertible on some nontrivial open interva/2[. Put
that X~(A)
QkCA):=
('
Uk(A)
0kCA)
0)(0
Xk(A)
2k(A)
.),
'
A e2[.
Then (V2) implies that Q~(A) decreases on 2[. Moreover, (V1) and (V~) imply that Qk(A)decreases strictly on 2[ provided k>~s holds, where~, E J is the strict contro//ab///ty index of
(HR).PROOF. Let k E J*\{0} and A E 2[. We may apply Lemma 4 with the conjoined basis
(X'(A),U*CA))
x(~)
~(~)
and
U'(~)=
U(~)
0(~)
of the"big" system from Lemma 4 so that for d E R 2n
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
Hamiitonian Eigenvalue P r o b l e m s
189
"* JCk(~)a= J{ O;(:,)x;-' (:,) - u;(:,)x;-' (~)x~(:,)x~ * - '--
{x:'/~>¢{x~"~>~-.:~>x~>}x~-'~>~~=o× x; - ' (:,)d
=-~'x;,-'(~,)d:~./
/xa+~(:,) T\ u~(~)
>(!00 o)}-em(~)0
(A)} d
A~(~)
o o
o
o
\ u;~(~)]
X,,,+~(A)~=o× X;-'k-1 ra=O Um
U~(~)
um(~) )
\
um(~)
(A)dT~rn
holds provided we assume (V2) and use the solution (x, u) of (HA) defined by
u~
:=
\ um(~) v~(~)
-
x~(~)
1), R~(~,)
d.
Now we assume (V 0 and (V2), let k> ss, and suppose dV(~(A)d= 0. This yields/:/re(A)\(x'~+lum) It follows that
= O,
for all0 _< m _< k - 1.
x'),o
x'),l .....
.. J
0
holds. Strict controllability of (HR) on J* with strict controllability index~s E J now forces x= u= 0 on J* so that d= 0 and hence Qk(A)< 0 follows.| LEMMA 6. (V1) and (V2) imply (I). PROOF. For every A E R, we denote the special normalized conjoined bases of (HA) at 0 by (X(A), U(A)) and (X(A), 0(A)). Let Ao E R. We pick a conjoined basis (.~, 0) of (HAo) such that (X(Ao), U(A0)) and (X, 0) are normalized and such that )(N+I is invertible (observe Lemma 1). Let (X(A), 0(A)) be the conjoined basis of (HA) with Xo(A)= )(o and 0o(A) -= 0o, A e R. Due to continuity, XN+I(A) is invertible on some nontrivial open interval that contains Ao, and on this interval we have strict monotonicity of I (-0N+I(A) 0 0 U~+~(,x)) (-RN+~(,X)XN+I(A),
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