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

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

2003-12-19

arxiv.org/abs/math/0312372v1

Mathematics

Spectral Theory

Scientific paper

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.

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