L2L2 spaces and boundary value problems on time-scales

In this paper we consider a second-order Sturm–Liouville-type boundary value operator of the formLu:=−[puΔΔ]+quσ,Lu:=−[puΔ]Δ+quσ, on an arbitrary, bounded time-scale TT, for suitable functions p,qp,q, together with suitable boundary conditions. Operators of this type on time-scales have normally been considered in a setting involving Banach spaces of continuous functions on TT. In this paper we introduce a space L2(T)L2(T) of square-integrable functions on TT, and Sobolev-type spaces Hn(T)Hn(T), n⩾1n⩾1, consisting of L2(T)L2(T) functions with n  th-order generalised L2(T)L2(T)-type derivatives. We prove some basic functional analytic results for these spaces, and then formulate the operator L   in this setting. In particular, we allow p∈H1(T)p∈H1(T), while q∈L2(T)q∈L2(T) — this generalises the usual conditions that p∈Crd1(Tκ), q∈Crd0(Tκ2). We give some immediate applications of the functional analytic results to L, such as ‘positivity’, injectivity, invertibility and compactness of the inverse. We also construct a Green's function for L. The analogues of these results on real intervals are well known, and are fundamental to the usual Sturm–Liouville theory on such intervals.