Optimal eigenvalues for some Laplacians and Schrödinger operators depending on curvature

1999/01/27 by Pavel Exner, Exner, Pavel, Evans M. Harrell +3
Mathematics · Physics and Astronomy · #FOS: Physical sciences #Mathematical Physics (math-ph) #Quantum Physics (quant-ph) #math-ph #math.MP #quant-ph

paper · pdf · doi:10.48550/arxiv.math-ph/9901022

Plain TeX, 11 pages; to appear in the Proceedings of QMath7, Birkhäuser Verlag, Basel 1999

arxiv created 1999/01/27 · arxiv updated 2009/11/30

Abstract

This article is an expanded version of the plenary talk given by Evans Harrell at QMath98, a meeting in Prague, June 1998. We consider Laplace operators and Schrödinger operators with potentials containing curvature on certain regions of nontrivial topology, especially closed curves, annular domains, and shells. Dirichlet boundary conditions are imposed on any boundaries. Under suitable assumptions we prove that the fundamental eigenvalue is maximized when the geometry is round. We also comment on the use of coordinate transformations for these operators and mention some open problems.

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