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Discrete spectrum of Schrödinger operators with potentials concentrated near conical surfaces

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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 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.

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Authors and Affiliations

  1. Department of Mathematics, Technion-Israel Institute of Technology, 629 Amado Building, 32000, Haifa, Israel

    Sebastian Egger

  2. Department of Mathematics and Computer Science, FernUniversität in Hagen, 58084, Hagen, Germany

    Joachim Kerner

  3. Département de mathématiques, Univ. Paris-Sud, CNRS, Université Paris-Saclay, 91405, Orsay Cedex, France

    Konstantin Pankrashkin

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  1. Sebastian Egger
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  2. Joachim Kerner
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  3. Konstantin Pankrashkin
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Correspondence to Joachim Kerner.

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Konstantin Pankrashkin: From March 1, 2020: Carl von Ossietzky Universität Oldenburg, Institut für Mathematik, 26111 Oldenburg, Germany.

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Egger, S., Kerner, J. & Pankrashkin, K. Discrete spectrum of Schrödinger operators with potentials concentrated near conical surfaces. Lett Math Phys 110, 945–968 (2020). https://doi.org/10.1007/s11005-019-01246-z

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  • DOI: https://doi.org/10.1007/s11005-019-01246-z

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