Evans M. Harrell
Commutators, eigenvalue gaps, and mean curvature in the theory of Schr\"odinger operators
(44K, latex)

ABSTRACT.  Commutator relations are used to investigate the spectra 
of Schr\"odinger Hamiltonians, $H = -\Delta + V\left({x}\right),$ 
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 
$\{\kappa_j\}$, $j = 1, \dots, d$ 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.