Electron. J. Diff. Eqns., Vol. 2007(2007), No. 175, pp. 1-10.

Self-adjoint boundary-value problems on time-scales

Fordyce A. Davidson, Bryan P. Rynne

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.

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Fordyce A. Davidson
Division of Mathematics, University of Dundee
Dundee, DD1 4HN, Scotland
email: fdavidso@maths.dundee.ac.uk
Bryan P. Rynne
Department of Mathematics and
the Maxwell Institute for Mathematical Sciences
Heriot-Watt University, Edinburgh EH14 4AS, Scotland
email: bryan@ma.hw.ac.uk

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