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Spectral optimization for strongly singular Schrödinger operators with a star-shaped interaction

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Abstract

We discuss the spectral properties of singular Schrödinger operators in three dimensions with the interaction supported by an equilateral star, finite or infinite. In the finite case, the discrete spectrum is nonempty if the star arms are long enough. Our main result concerns spectral optimization: we show that the principal eigenvalue is uniquely maximized when the arms are arranged in one of the known five sharp configurations.

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Acknowledgements

The research was supported by the Czech Science Foundation (GAČR) under Grant No. 17-01706S and by the European Union within the project CZ.02.1.01/0.0/0.0/16 019/0000778. S.K. also acknowledges the financial support from the Polish ‘Regional Initiative of Excellence 2019-2022’ Program, Project No. 003/RID/2018/19. Thanks also go to the referees for a careful reading of the manuscript and their useful remarks and comments.

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Authors and Affiliations

  1. Doppler Institute for Mathematical Physics and Applied Mathematics, Czech Technical University in Prague, Břehová 7, 11519, Prague, Czechia

    P. Exner

  2. Nuclear Physics Institute CAS, 25068, Řež, Prague, Czech Republic

    P. Exner

  3. Institute of Physics, University of Zielona Góra, ul. Szafrana 4a, 65246, Zielona Gora, Poland

    S. Kondej

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  1. P. Exner
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  2. S. Kondej
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Correspondence to S. Kondej.

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Exner, P., Kondej, S. Spectral optimization for strongly singular Schrödinger operators with a star-shaped interaction. Lett Math Phys 110, 735–751 (2020). https://doi.org/10.1007/s11005-019-01237-0

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