Attractive conical surfaces create infinitely many bound states

Abstract

In this paper we study spectral properties of a three-dimensional Schrödinger 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.

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Cited by in corpus (2)

• Spectral properties of soft quantum waveguides
• Soft quantum waveguides in three dimensions