Self-adjoint boundary-value problems on time-scales

NOTE: THE MATHEMATICAL SYMBOLS IN THIS ABSTRACT CANNOT BE DISPLAYED CORRECTLY ON THIS PAGE. PLEASE REFER TO THE ABSTRACT IN THE ATTACHED FILE OR THE PUBLISHERS WEBSITE FOR AN ACCURATE DISPLAY. In this paper we consider a second order, Sturm-Liouville-type boundary-value operator of the form Lu := -[pur] + 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 L2(T ), 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.