Davidson, Fordyce A.; Rynne, Bryan P. Self-adjoint boundary-value problems on time-scales. (English) Zbl 1138.34004 Electron. J. Differ. Equ. 2007, Paper No. 175, 10 p. (2007). Summary: 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. Cited in 5 Documents MSC: 34B05 Linear boundary value problems for ordinary differential equations 39A05 General theory of difference equations 47E05 General theory of ordinary differential operators 34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.) Keywords:Time-scales; boundary-value problem; self-adjoint linear operators; Sobolev spaces PDFBibTeX XMLCite \textit{F. A. Davidson} and \textit{B. P. Rynne}, Electron. J. Differ. Equ. 2007, Paper No. 175, 10 p. (2007; Zbl 1138.34004) Full Text: EuDML EMIS