Letters in Mathematical Physics

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

A spectral isoperimetric inequality for cones

  • Pavel Exner
  • Vladimir Lotoreichik


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.


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

Mathematics Subject Classification

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



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.


  1. 1.
    Abrams, A., Cantarella, J., Fu, J.H., Ghomi, M., Howard, R.: Circles minimize most knot energies. Topology 42, 381–394 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 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)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. 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)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Behrndt, J., Exner, P., Lotoreichik, V.: Schrödinger operators with \(\delta \)-interactions supported on conical surfaces. J. Phys. A Math. Theor. 47, 355202 (2014)CrossRefzbMATHGoogle Scholar
  5. 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. 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)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Birman, M.Sh., Solomjak, M.Z.: Spectral Theory of Self-Adjoint Operators in Hilbert Spaces. D. Reidel Publishing Co., Dordrecht (1987)Google Scholar
  8. 8.
    Blank, J., Exner, P., Havlíček, M.: Hilbert Space Operators in Quantum Physics. Theoretical and Mathematical Physics. Springer, Berlin (2008)zbMATHGoogle Scholar
  9. 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)Google Scholar
  10. 10.
    Brasche, J.F., Exner, P., Kuperin, Y.A., Šeba, P.: Schrödinger operators with singular interactions. J. Math. Anal. Appl. 184, 112–139 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Bruneau, V., Popoff, N.: On the negative spectrum of the Robin Laplacian in corner domains. Anal. PDE 9, 1259–1283 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Daners, D.: A Faber–Krahn inequality for Robin problems in any space dimension. Math. Ann. 335, 767–785 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Duclos, P., Exner, P., Krejčiřík, D.: Bound states in curved quantum layers. Commun. Math. Phys. 223, 13–28 (2001)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Exner, P.: An isoperimetric problem for point interactions. J. Phys. A Math. Gen. 38, 4795–4802 (2005)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Exner, P.: Necklaces with interacting beads: isoperimetric problems. Contemp. Math. 412, 141–149 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 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)Google Scholar
  17. 17.
    Exner, P., Fraas, M.: On geometric perturbations of critical Schrödinger operators with a surface interaction. J. Math. Phys. 50, 112101 (2009)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  18. 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)ADSCrossRefzbMATHGoogle Scholar
  19. 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)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Exner, P., Kovařík, H.: Quantum Waveguides. Theoretical and Mathematical Physics. Springer, Cham (2015)CrossRefzbMATHGoogle Scholar
  21. 21.
    Exner, P., Rohleder, J.: Generalized interactions supported on hypersurfaces. J. Math. Phys. 57, 041507 (2016)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  22. 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)Google Scholar
  23. 23.
    Freitas, P., Krejčiřík, D.: The first Robin eigenvalue with negative boundary parameter. Adv. Math. 280, 322–339 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Goldberg, M.: Dispersive estimates for Schrödinger operators with measure-valued potentials in \({\mathbb{R}}^{3}\). Indiana Univ. Math. J. 61, 2123–2141 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Helffer, B.: Spectral Theory and Its Applications. Cambridge University Press, Cambridge (2013)zbMATHGoogle Scholar
  26. 26.
    Kato, T.: Perturbation Theory for Linear Operators. Classics in Mathematics. Springer, Berlin (1995)Google Scholar
  27. 27.
    Krahn, E.: Über eine von Rayleigh formulierte Minimaleigenschaft des Kreises. Math. Ann. 94, 97–100 (1925)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Levitin, M., Parnovski, L.: On the principal eigenvalue of a Robin problem with a large parameter. Math. Nachr. 281, 272–281 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 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)CrossRefzbMATHGoogle Scholar
  30. 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)CrossRefzbMATHGoogle Scholar
  31. 31.
    Lükő, G.: On the mean length of the chords of a closed curve. Isr. J. Math. 4, 23–32 (1966)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    O’Hara, J.: Energy of Knots and Conformal Geometry. World Scientific, Singapore (2003)CrossRefzbMATHGoogle Scholar
  33. 33.
    Pankrashkin, K.: On the discrete spectrum of Robin Laplacians in conical domains. Math. Model. Nat. Phenom. 11, 100–110 (2016)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics. I: Functional Analysis. Rev. and enl. ed. Academic Press, New York (1980)zbMATHGoogle Scholar
  35. 35.
    Teschl, G.: Mathematical Methods in Quantum Mechanics. With Applications to Schrödinger Operators. Graduate Studies in Mathematics. American Mathematical Society, Providence (2014)zbMATHGoogle Scholar
  36. 36.
    Weidmann, J.: Lineare Operatoren in Hilberträumen: Teil 1 Grundlagen. Teubner, Wiesbaden (2000)CrossRefGoogle Scholar

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© 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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