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Spectral Isoperimetric Inequality for the δ′-Interaction on a Contour

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Mathematical Challenges of Zero-Range Physics

Part of the book series: Springer INdAM Series ((SINDAMS,volume 42))

Abstract

We consider the problem of geometric optimization for the lowest eigenvalue of the two-dimensional Schrödinger operator with an attractive δ′-interaction of a fixed strength, the support of which is a C 2-smooth contour. Under the constraint of a fixed length of the contour, we prove that the lowest eigenvalue is maximized by the circle. The proof relies on the min-max principle and the method of parallel coordinates.

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Acknowledgements

The author is indebted to Pavel Exner, Michal Jex, David Krejčiřík, and Magda Khalile for fruitful discussions and gratefully acknowledges the support by the grant No. 17-01706S of the Czech Science Foundation (GAČR).

The author also thanks Alessandro Michelangeli for the invitation to participate in and give a mini-course at the third workshop: Mathematical Challenges of Zero-Range Physics: rigorous results and open problems and Istituto Nazionale di Alta Matematica “Francesco Severi” for the financial support of the travel.

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Authors and Affiliations

  1. Department of Theoretical Physics, Nuclear Physics Institute, Czech Academy of Sciences, Řež, Czech Republic

    Vladimir Lotoreichik

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  1. Vladimir Lotoreichik
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Correspondence to Vladimir Lotoreichik .

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Editors and Affiliations

  1. Institute for Applied Mathematics, and Hausdorff Center for Mathematics, University of Bonn, Bonn, Germany

    Alessandro Michelangeli

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Lotoreichik, V. (2021). Spectral Isoperimetric Inequality for the δ′-Interaction on a Contour. In: Michelangeli, A. (eds) Mathematical Challenges of Zero-Range Physics. Springer INdAM Series, vol 42. Springer, Cham. https://doi.org/10.1007/978-3-030-60453-0_10

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