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Self-adjoint boundary-value problems on time-scales. (English) Zbl 1138.34004

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.

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.)
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