Electronic Journal of Differential Equations,
Vol. 2007(2007), No. 175, pp. 1-10.

Title: Self-adjoint boundary-value problems on time-scales

Authors: Fordyce A. Davidson (Dundee University, Scotland, UK)
         Bryan P. Rynne      (Dundee University, Scotland, UK)

Abstract: 
 In this paper we consider a second order, Sturm-Liouville-type
 boundary-value operator of the form
 $$
 L u := -[p u^{\nabla}]^{\Delta} + qu,
 $$
 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.
 
Submitted June 6, 2007. Published December 12, 2007.
Math Subject Classifications: 34B05, 34L05, 39A05.
Key Words: Time-scales; boundary-value problem; 
           self-adjoint linear operators; Sobolev spaces.