A continuum of unusual self-adjoint linear partial differential operators

Abstract

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 Ω 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 ( Ω ) ; but here for brevity we restrict the development to the classical Laplacian differential expression, with Ω now the planar unit disk. It is demonstrated that there exists a non-denumerable set of such T Har -like operators (each a self-adjoint extension generated by the Laplacian), each of which has a domain in L 2 ( Ω ) that does not lie within the usual Sobolev Hilbert function space W 2 ( Ω ) . These T Har -like operators cannot be specified by conventional differential boundary conditions on the boundary of ∂ Ω , and may have non-empty essential spectra.