**
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.