A continuum of unusual self-adjoint linear partial differential operators

In an earlier publication a linear operator THar 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 THar-like self-adjoint operators on the Hilbert function space L2([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 L2([Omega]) that does not lie within the usual Sobolev Hilbert function space W2([Omega]). These THar-like operators cannot be specified by conventional differential boundary conditions on the boundary of [partial differential][Omega], and may have non-empty essential spectra.