Small perturbations and the eigenvalues of the Laplacian on large bounded domains

Abstract:

Denote by $\Delta _L^D$ the Laplacian on a hypercube in ${{\mathbf {R}}^d}$ with side length $\pi L$. Also denote by $N\left ( {\lambda ,A} \right )$ the number of eigenvalues of the operator $A$ below $\lambda$. If $V \geq 0$ is a bounded function of compact support, ($V > 0$ on a set of positive measure) then $N\left ( { - \Delta _L^D,\lambda } \right ) - N\left ( { - \Delta _L^D + V,\lambda } \right )$ is not bounded as $L \to \infty$ for dimension $d > 1$.