Commutators, eigenvalue gaps, and mean curvature in the theory of Schrödinger operators

Abstract

Commutator relations are used to investigate the spectra of Schrödinger Hamiltonians, $H = -Δ+ V({x}),$ acting on functions of a smooth, compact $d$-dimensional manifold $M$ immersed in $\bbr^ν, ν\geq d+1$. Here $Δ$ 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: $$H_{g} := -Δ+ g h^2,$$ where $ν= d+1$, $g$ is a real parameter, and $$h := \sum\limits_{j = 1}^{d} {κ_j},$$ with $\{κ_j\}$, $j = 1, ..., d$ denoting the principal curvatures of $M$. For instance, by Theorem \ref{thm3.1} and Corollary \ref{cor4.5}, 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.