A continuum of unusual selfadjoint linear partial differential operators
Abstract
In an earlier publication a linear operator THar was defined as an unusual selfadjoint 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 THarlike selfadjoint 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 nondenumerable set of such THarlike operators (each a selfadjoint 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 THarlike operators cannot be specified by conventional differential boundary conditions on the boundary of [partial differential][Omega], and may have nonempty essential spectra.
 Publication:

Journal of Computational and Applied Mathematics
 Pub Date:
 November 2007
 Bibcode:
 2007JCoAM.208..164E
 Keywords:

 Linear partial differential equations;
 Selfadjoint partial differential equations;
 Spectral theory