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