Optimal Eigenvalues for Some Laplacians and Schrödinger Operators Depending on Curvature
Pavel Exner, Evans M. Harrell, Michael Loss · Birkhäuser Basel eBooks · 1999
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.