A continuum of unusual self-adjoint linear partial differential operators

In an earlier publication a linear operator T-Har was defined as an unusual self-adjoint extension generated by each linear elliptic partial differential expression, satisfying suitable conditions on a bounded region Omega of some Euclidean space. In this present work the authors define an extensive class of T-Har-like self-adjoint operators on the Hilbert function space L-2(Omega); but here for brevity we restrict the development to the classical Laplacian differential expression, with Omega now the planar unit disk. It is demonstrated that there exists a non-denumerable set of such THar-like operators (each a self-adjoint extension generated by the Laplacian), each of which has a domain in L-2(Omega) that does not lie within the usual Sobolev Hilbert function space W-2(Omega). These THar-like operators cannot be specified by conventional differential boundary conditions on the boundary of 80, and may have non-empty essential spectra. (c) 2006 Elsevier B.V. All rights reserved.