Below is the ascii version of the abstract for 03-546. The html version should be ready soon.

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 
$$h := \sum\limits_{j = 1}^{d} {\kappa_j},$$ 
$\{\kappa_j\}$, $j = 1, \dots, d$ 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.