99-32 Exner P., Harrell E.M., Loss M.

Optimal eigenvalues for some Laplacians and Schr\"odinger operators depending on curvature. (34K, plain TeX) Jan 26, 99

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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 Schr\"odinger 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.

Files: 99-32.tex