A continuum of unusual self-adjoint linear partial differential operators |
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Authors: | W.N. Everitt L. Markus M. Muzzulini M. Plum |
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Affiliation: | 1. School of Mathematics, University of Birmingham, Edgbaston, Birmingham B15 2TT, England, UK;2. School of Mathematics, University of Minnesota, Minneapolis, MN 55455-0487, USA;3. Mathematisches Institut I, Universität Karlsruhe, D-76128, Germany |
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Abstract: | 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 Ω 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(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 THar-like operators (each a self-adjoint extension generated by the Laplacian), each of which has a domain in L2(Ω) that does not lie within the usual Sobolev Hilbert function space W2(Ω). These THar-like operators cannot be specified by conventional differential boundary conditions on the boundary of ∂Ω, and may have non-empty essential spectra. |
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Keywords: | primary, 35J40, 35J67, 35P05 secondary, 32A36, 32A40, 47B25 |
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