Heriot-Watt Mathematics Report Series
HWM07-18, 2 Jul 2007
Eigenfunction expansions for boundary value problems on time-scales
B P Rynne and F A Davidson
Abstract
Let $\T \subset \R$ be a bounded time-scale,
with $a = \inf\T $, $b = \sup \T$.
We consider the weighted, linear, eigenvalue problem
\begin{gather*}
-(p u^{\De})^{\De}(t) +q(t)u^\si(t) = \la w(t) u^\si(t) ,
\quad t \in \T^{\ka^2},
%\label{eval_prob.eq}
\\
c_{00} u(a) + c_{01} u^\De(a) = 0,
\quad
c_{10} u(\rho(b)) + c_{11} u^\De(\rho(b)) = 0,
%\label{bc.eq}
\end{gather*}
for suitable functions $p,\,q$ and $w$ and $\la \in \R$.
Problems of this type on time-scales have normally been considered
in a setting involving Banach spaces of
continuous functions on $\T$.
In this paper we formulate the problem in Sobolev-type spaces
of functions with generalised $L^2$-type derivatives.
This approach allows us to use the functional analytic theory of
Hilbert spaces rather than Banach spaces.
Moreover, it allows us to
use more general coefficient
functions $p,\,q$, and weight function $w$, than usual,
viz., $p \in H^1(\T^\ka)$ and $q,\, w \in L^2(\T^\ka)$
compared with the usual hypotheses that
$p \in \Crd^1(\T^\ka)$, $q,\, w \in \Crd^0(\T^{\ka^2})$.
Further to these conditions, we assume that
$p \ge c >0$, on $\T^\ka$, $C \ge w \ge c > 0$ on $\T^{\ka^2}$,
for some constants $C > c > 0$.
These conditions are similar to the usual assumptions imposed
on Sturm-Liouville, ordinary differential equation problems.
We obtain a min-max characterization of the eigenvalues of the above problem,
and various eigenfunction expansions for functions in suitable
function spaces.
These results extend certain aspects of the standard theory of self-adjoint
operators with compact resolvent to the above problem,
even though the linear operators associated with the left hand side of
the problem is not in fact self-adjoint on general time-scales.
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