Self-adjoint boundary-value problems on time-scales

Abstract

In this paper we consider a second order, Sturm-Liouville-type boundary-value operator of the form $$L u := -[p u^{ abla}]^{Delta} + qu,$$ on an arbitrary, bounded time-scale $mathbb{T}$, 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 $L^2(mathbb{T}_kappa)$, 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