Skip to main content
SpringerLink
Log in

A spectral isoperimetric inequality for cones

  • Published:
Letters in Mathematical Physics Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abrams, A., Cantarella, J., Fu, J.H., Ghomi, M., Howard, R.: Circles minimize most knot energies. Topology 42, 381–394 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  2. Arrizabalaga, N., Mas, A., Vega, L.: An isoperimetric-type inequality for electrostatic shell interactions for Dirac operators. Commun. Math. Phys. 344, 483–505 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  3. Behrndt, J., Exner, P., Lotoreichik, V.: Schrödinger operators with \(\delta \)- and \(\delta ^{\prime }\)-interactions on Lipschitz surfaces and chromatic numbers of associated partitions. Rev. Math. Phys. 26, 1450015 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  4. Behrndt, J., Exner, P., Lotoreichik, V.: Schrödinger operators with \(\delta \)-interactions supported on conical surfaces. J. Phys. A Math. Theor. 47, 355202 (2014)

    Article  MATH  Google Scholar 

  5. Behrndt, J., Frank, R.L., Kühn, C., Lotoreichik, V., Rohleder, J.: Spectral theory for Schrödinger operators with \(\delta \)-interactions supported on curves in \({\mathbb{R}^3}\). Ann. Henri Poincaré (To appear). arXiv:1601.06433

  6. Behrndt, J., Langer, M., Lotoreichik, V.: Schrödinger operators with \(\delta \) and \(\delta ^{\prime }\)-potentials supported on hypersurfaces. Ann. Henri Poincaré 14, 385–423 (2013)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  7. Birman, M.Sh., Solomjak, M.Z.: Spectral Theory of Self-Adjoint Operators in Hilbert Spaces. D. Reidel Publishing Co., Dordrecht (1987)

  8. Blank, J., Exner, P., Havlíček, M.: Hilbert Space Operators in Quantum Physics. Theoretical and Mathematical Physics. Springer, Berlin (2008)

    MATH  Google Scholar 

  9. Brasche, J.F.: On the spectral properties of singularly perturbed operators. In: Zhiming, M., Röckner, J.,Yan, J.A. (eds.) Dirichlet Forms and Stochastic Processes, pp. 65–72. de Gruyter (1995)

  10. Brasche, J.F., Exner, P., Kuperin, Y.A., Šeba, P.: Schrödinger operators with singular interactions. J. Math. Anal. Appl. 184, 112–139 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  11. Bruneau, V., Popoff, N.: On the negative spectrum of the Robin Laplacian in corner domains. Anal. PDE 9, 1259–1283 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  12. Daners, D.: A Faber–Krahn inequality for Robin problems in any space dimension. Math. Ann. 335, 767–785 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  13. Duclos, P., Exner, P., Krejčiřík, D.: Bound states in curved quantum layers. Commun. Math. Phys. 223, 13–28 (2001)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  14. Exner, P.: An isoperimetric problem for point interactions. J. Phys. A Math. Gen. 38, 4795–4802 (2005)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  15. Exner, P.: Necklaces with interacting beads: isoperimetric problems. Contemp. Math. 412, 141–149 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  16. Exner, P.: Leaky quantum graphs: a review. In: Analysis on Graphs and Its Applications. Selected papers based on the Isaac Newton Institute for Mathematical Sciences programme, Cambridge, UK, 2007. Proc. Symp. Pure Math., vol. 77, pp. 523–564 (2008)

  17. Exner, P., Fraas, M.: On geometric perturbations of critical Schrödinger operators with a surface interaction. J. Math. Phys. 50, 112101 (2009)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  18. Exner, P., Fraas, M., Harrell, E.M.: On the critical exponent in an isoperimetric inequality for chords. Phys. Lett. A 368, 1–6 (2007)

    Article  ADS  MATH  Google Scholar 

  19. Exner, P., Harrell, E.M., Loss, M.: Inequalities for means of chords, with application to isoperimetric problems. Lett. Math. Phys. 75, 225–233 (2006)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  20. Exner, P., Kovařík, H.: Quantum Waveguides. Theoretical and Mathematical Physics. Springer, Cham (2015)

    Book  MATH  Google Scholar 

  21. Exner, P., Rohleder, J.: Generalized interactions supported on hypersurfaces. J. Math. Phys. 57, 041507 (2016)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  22. Faber, G.: Beweis, dass unter allen homogenen Membranen von gleicher Fläche und gleicher Spannung die kreisförmige den tiefsten Grundton gibt. Verlagd. Bayer. Akad. d. Wiss. (1923)

  23. Freitas, P., Krejčiřík, D.: The first Robin eigenvalue with negative boundary parameter. Adv. Math. 280, 322–339 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  24. Goldberg, M.: Dispersive estimates for Schrödinger operators with measure-valued potentials in \({\mathbb{R}}^{3}\). Indiana Univ. Math. J. 61, 2123–2141 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  25. Helffer, B.: Spectral Theory and Its Applications. Cambridge University Press, Cambridge (2013)

    MATH  Google Scholar 

  26. Kato, T.: Perturbation Theory for Linear Operators. Classics in Mathematics. Springer, Berlin (1995)

    Google Scholar 

  27. Krahn, E.: Über eine von Rayleigh formulierte Minimaleigenschaft des Kreises. Math. Ann. 94, 97–100 (1925)

    Article  MathSciNet  MATH  Google Scholar 

  28. Levitin, M., Parnovski, L.: On the principal eigenvalue of a Robin problem with a large parameter. Math. Nachr. 281, 272–281 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  29. Lotoreichik, V., Ourmières-Bonafos, T.: On the bound states of Schrödinger operators with \(\delta \)-interactions on conical surfaces. Commun. Partial Differ. Equ. 41, 999–1028 (2016)

    Article  MATH  Google Scholar 

  30. Lotoreichik, V., Rohleder, J.: An eigenvalue inequality for Schrödinger operators with \(\delta \)- and \(\delta ^\prime \)-interactions supported on hypersurfaces. Oper. Theory Adv. Appl. 247, 173–184 (2015)

    Article  MATH  Google Scholar 

  31. Lükő, G.: On the mean length of the chords of a closed curve. Isr. J. Math. 4, 23–32 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  32. O’Hara, J.: Energy of Knots and Conformal Geometry. World Scientific, Singapore (2003)

    Book  MATH  Google Scholar 

  33. Pankrashkin, K.: On the discrete spectrum of Robin Laplacians in conical domains. Math. Model. Nat. Phenom. 11, 100–110 (2016)

    Article  MathSciNet  Google Scholar 

  34. Reed, M., Simon, B.: Methods of Modern Mathematical Physics. I: Functional Analysis. Rev. and enl. ed. Academic Press, New York (1980)

    MATH  Google Scholar 

  35. Teschl, G.: Mathematical Methods in Quantum Mechanics. With Applications to Schrödinger Operators. Graduate Studies in Mathematics. American Mathematical Society, Providence (2014)

    MATH  Google Scholar 

  36. Weidmann, J.: Lineare Operatoren in Hilberträumen: Teil 1 Grundlagen. Teubner, Wiesbaden (2000)

    Book  Google Scholar 

Download references

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.

Author information

Authors and Affiliations

  1. Department of Theoretical Physics, Nuclear Physics Institute, Czech Academy of Sciences, 25068, Řež near Prague, Czech Republic

    Pavel Exner & Vladimir Lotoreichik

  2. Doppler Institute for Mathematical Physics and Applied Mathematics, Czech Technical University, Břehová 7, 11519, Prague, Czech Republic

    Pavel Exner

Authors
  1. Pavel Exner
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Vladimir Lotoreichik
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Pavel Exner.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Exner, P., Lotoreichik, V. A spectral isoperimetric inequality for cones. Lett Math Phys 107, 717–732 (2017). https://doi.org/10.1007/s11005-016-0917-8

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11005-016-0917-8

Keywords

Mathematics Subject Classification

Search

Navigation