Commutators, eigenvalue gaps, and mean curvature in the theory of Schr\"odinger operators

Abstract

Commutator relations are used to investigate the spectra of Schr\"odinger Hamiltonians, $H = -\Delta + V({x}),$ acting on functions of a smooth, compact $d$-dimensional manifold $M$ immersed in $\bbr^{\nu}, \nu \geq d+1$. Here $\Delta$ 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 r\^ole is found to be played by a Schr\"odinger operator with potential proportional to the square of the mean curvature: $$H_{g} := -\Delta + g h^2,$$ where $\nu = d+1$, $g$ is a real parameter, and $$h := \sum\limits_{j = 1}^{d} {\kappa_j},$$ with $\{\kappa_j\}$, $j = 1, ..., d$ denoting the principal curvatures of $M$. For instance, by Theorem \ref{thm3.1} and Corollary \ref{cor4.5}, each eigenvalue gap of an arbitrary Schr\"odinger 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.