Abstract

Commutator relations are used to investigate the spectra of Schrödinger Hamiltonians, H = − ∆ + V (x) , acting on functions of a smooth, compact d-dimensional manifold M immersed in R d+1. Here ∆ denotes the Laplace-Beltrami operator, and the real-valued potential–energy function V (x) acts by multiplication. The manifold M may be complete or it may have a boundary, in which case Dirichlet boundary conditions are imposed. It is found that the mean curvature of a manifold poses tight constraints on the spectrum of H. Further, a special algebraic rôle is found to be played by a Schrödinger operator with potential proportional to the square of the mean curvature: Hg: = − ∆ + gh 2, where g is a real parameter and h:= d� κj, j=1 with {κj}, j = 1,..., d denoting the principal curvatures of M. For instance, by Theorem 2.1 and Corollary 3.4, each eigenvalue gap of an arbitrary Schrödinger operator is bounded above by an expression using H 1/4. The “isoperimetric ” parts of these theorems state that these bounds are sharp for the fundamental eigenvalue gap and for infinitely many other eigenvalue gaps.

Citations