Discrete spectrum of Schrödinger operators with potentials concentrated near conical surfaces
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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 to an infinite conical surface 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.
KeywordsDiscrete spectrum Conical surface Attractive potential
Mathematics Subject Classification81Q10 81Q80 81Q37 81Q35
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