Discrete spectrum of Schrödinger operators with potentials concentrated near conical surfaces

Abstract

In this paper we study spectral properties of a three-dimensional Schr\"odinger operator $-\Delta+V$ with a potential $V$ given, modulo rapidly decaying terms, by a function of the distance of $x \in \mathbb{R}^3$ to an infinite conical hypersurface with a smooth cross-section. As a main result we show that there are infinitely many discrete eigenvalues accumulating at the bottom of the essential spectrum which itself is identified as the ground-state energy of a certain one-dimensional operator. Most importantly, based on a result of Kirsch and Simon we are able to establish the asymptotic behavior of the eigenvalue counting function using an explicit spectral-geometric quantity associated with the cross-section. This shows a universal character of some previous results on conical layers and $\delta$-potentials created by conical surfaces.