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Letters in Mathematical Physics

, Volume 107, Issue 4, pp 717–732 | Cite as

A spectral isoperimetric inequality for cones

  • Pavel Exner
  • Vladimir Lotoreichik
Article

Abstract

In this note, we investigate three-dimensional Schrödinger operators with \(\delta \)-interactions supported on \(C^2\)-smooth cones, both finite and infinite. Our main results concern a Faber–Krahn-type inequality for the principal eigenvalue of these operators. The proofs rely on the Birman-Schwinger principle and on the fact that circles are unique minimizers for a class of energy functionals. The main novel idea consists in the way of constructing test functions for the Birman-Schwinger principle.

Keywords

Schrödinger operator \(\delta \)-interaction Conical surface Isoperimetric inequality Existence of bound states 

Mathematics Subject Classification

Primary 35P15 Secondary 35J10 35Q40 46F10 81Q10 81Q37 

Notes

Acknowledgements

This research was supported by the Czech Science Foundation (GAČR) within the project 14-06818S. We are grateful to the anonymous referee, whose suggestion inspired Remark 1.4.

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Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Department of Theoretical Physics, Nuclear Physics InstituteCzech Academy of SciencesŘež near PragueCzech Republic
  2. 2.Doppler Institute for Mathematical Physics and Applied MathematicsCzech Technical UniversityPragueCzech Republic

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