Spectral inequality for Schrödinger's equation with multipoint potential

Abstract

Schrödinger's equation with potential that is a sum of a regular function and a finite set of point scatterers of Bethe-Peierls type is under consideration. For this equation the spectral problem with homogeneous linear boundary conditions is considered, which covers the Dirichlet, Neumann, and Robin cases. It is shown that when the energy $E$ is an eigenvalue with multiplicity $m$, it remains an eigenvalue with multiplicity at least $m-n$ after adding $n<m$ point scatterers. As a consequence, because for the zero potential all values of the energy are transmission eigenvalues with infinite multiplicity, this property also holds for $n$-point potentials, as discovered originally in a recent paper by the authors. Bibliography: 33 titles.