Commutators, eigenvalue gaps, and mean curvature in the theory of Schr\"odinger operators

Commutator relations are used to investigate the spectra of Schr"odinger Hamiltonians, $H=-\mathrm{\Delta }+V(x),$ acting on functions of a smooth, compact $d$-dimensional manifold $M$ immersed in ${\text{bbr}}^{\nu },\nu \ge d+1$. Here $\mathrm{\Delta }$ 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: $$H_g:=-\mathrm{\Delta }+g{h}^2,$$ where $\nu =d+1$, $g$ is a real parameter, and $$h:=\sum _{j=1}^d{\kappa }_j,$$ with $\{{\kappa }_j\}$, $j=1,\mathrm{.}\mathrm{.}\mathrm{.},d$ 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 $H_{1\mathrm{/}4}$. The “isoperimetric” parts of these theorems state that these bounds are sharp for the fundamental eigenvalue gap and for infinitely many other eigenvalue gaps.