Heriot-Watt Mathematics Report Series
HWM07-21, 2 Jul 2007
Self-adjoint boundary value problems on time-scales
B P Rynne and F A Davidson
Abstract
In this paper we consider a
second order, Sturm-Liouville-type boundary value operator
of the form
$$
L u := -[p u^{\na}]^{\De} + qu,
$$
on an arbitrary, bounded time-scale $\T$, for suitable functions $p,\,q$,
together with suitable boundary conditions.
We show that, with a suitable choice of domain, this operator can be formulated in
the Hilbert space $L^2(\T_\ka)$,
in such a way that the resulting operator is self-adjoint,
with compact resolvent
(here, `self-adjoint' means in the standard functional analytic meaning of this term).
Previous discussions of operators of this, and similar, form have described them as
`self-adjoint', but have not demonstrated self-adjointness in the
standard functional analytic sense.
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