Skip to main content
SpringerLink
Log in

Leaky Quantum Structures

  • Research Articles
  • Published:
Proceedings of the Steklov Institute of Mathematics Aims and scope Submit manuscript

Abstract

The paper reviews spectral properties of a class of singular Schrödinger operators with the interaction supported by manifolds or complexes of codimension \(1\); in particular, the relation of these properties to the geometric setting of the problem is discussed. We describe how these operators can be approximated by operators of other classes and how such approximations can be used. Furthermore, we present asymptotic expansions of the eigenvalues in terms of the parameters characterizing the coupling strength and geometric deformations. We also give an example illustrating the influence of a magnetic field of the Aharonov–Bohm type and briefly describe results on singular perturbations of Dirac operators.

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

Fig. 1.
Fig. 2.
Fig. 3.
Fig. 4.
Fig. 5.

Similar content being viewed by others

References

  1. R. Adami and A. Teta, “On the Aharonov–Bohm Hamiltonian,” Lett. Math. Phys. 43 (1), 43–54 (1998).

    Article  MathSciNet  MATH  Google Scholar 

  2. S. Albeverio, F. Gesztesy, R. Høegh-Krohn, and H. Holden, Solvable Models in Quantum Mechanics, 2nd ed. (AMS Chelsea Publ., Providence, RI, 2005).

    MATH  Google Scholar 

  3. J.-P. Antoine, F. Gesztesy, and J. Shabani, “Exactly solvable models of sphere interactions in quantum mechanics,” J. Phys. A: Math. Gen. 20 (12), 3687–3712 (1987).

    Article  MathSciNet  Google Scholar 

  4. N. Arrizabalaga, L. Le Treust, and N. Raymond, “On the MIT bag model in the non-relativistic limit,” Commun. Math. Phys. 354 (2), 641–669 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  5. N. Arrizabalaga, A. Mas, and L. Vega, “Shell interactions for Dirac operators,” J. Math. Pures Appl. 102 (4), 617–639 (2014).

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  7. J. Behrndt, P. Exner, M. Holzmann, and V. Lotoreichik, “Approximation of Schrödinger operators with \(\delta \) -interactions supported on hypersurfaces,” Math. Nachr. 290 (8–9), 1215–1248 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  8. J. Behrndt, P. Exner, M. Holzmann, and V. Lotoreichik, “On the spectral properties of Dirac operators with electrostatic \(\delta \)-shell interactions,” J. Math. Pures Appl. 111, 47–78 (2018).

    Article  MathSciNet  MATH  Google Scholar 

  9. J. Behrndt, P. Exner, M. Holzmann, and V. Lotoreichik, “On Dirac operators in \(\mathbb R^3\) with electrostatic and Lorentz scalar \(\delta \)-shell interactions,” Quantum Stud. Math. Found. 6 (3), 295–314 (2019).

    Article  MathSciNet  MATH  Google Scholar 

  10. J. Behrndt, P. Exner, and V. Lotoreichik, “Schrödinger operators with \(\delta \)- and \(\delta '\)-interactions on Lipschitz surfaces and chromatic numbers of associated partitions,” Rev. Math. Phys. 26 (8), 1450015 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  11. J. Behrndt, P. Exner, and V. Lotoreichik, “Schrödinger operators with \(\delta \)-interactions supported on conical surfaces,” J. Phys. A: Math. Theor. 47 (35), 355202 (2014).

    Article  MATH  Google Scholar 

  12. J. Behrndt and M. Holzmann, “On Dirac operators with electrostatic \(\delta \)-shell interactions of critical strength,” J. Spectr. Theory 10 (1), 147–184 (2020).

    Article  MathSciNet  MATH  Google Scholar 

  13. J. Behrndt, M. Holzmann, T. Ourmières-Bonafos, and K. Pankrashkin, “Two-dimensional Dirac operators with singular interactions supported on closed curves,” J. Funct. Anal. 279 (8), 108700 (2020); arXiv: 1907.05436 [math.AP].

    Article  MathSciNet  MATH  Google Scholar 

  14. R. D. Benguria, S. Fournais, E. Stockmeyer, and H. Van Den Bosch, “Self-adjointness of two-dimensional Dirac operators on domains,” Ann. Henri Poincaré 18 (4), 1371–1383 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  15. R. D. Benguria, S. Fournais, E. Stockmeyer, and H. Van Den Bosch, “Spectral gaps of Dirac operators describing graphene quantum dots,” Math. Phys. Anal. Geom. 20 (2), 11 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  16. J. F. Brasche, P. Exner, Yu. A. Kuperin, and P. Šeba, “Schrödinger operators with singular interactions,” J. Math. Anal. Appl. 184 (1), 112–139 (1994).

    Article  MathSciNet  MATH  Google Scholar 

  17. J. F. Brasche, R. Figari, and A. Teta, “Singular Schrödinger operators as limits of point interaction Hamiltonians,” Potential Anal. 8 (2), 163–178 (1998).

    Article  MathSciNet  MATH  Google Scholar 

  18. J. F. Brasche and K. Ožanová, “Convergence of Schrödinger operators,” SIAM J. Math. Anal. 39 (1), 281–297 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  19. B. M. Brown, M. S. P. Eastham, A. M. Hinz, T. Kriecherbauer, D. K. R. McCormack, and K. M. Schmidt, “Welsh eigenvalues of radially periodic Schrödinger operators,” J. Math. Anal. Appl. 225 (1), 347–357 (1998).

    Article  MathSciNet  MATH  Google Scholar 

  20. L. Dąbrowski and P. Šťovíček, “Aharonov–Bohm effect with \(\delta \)-type interaction,” J. Math. Phys. 39 (1), 47–62 (1998).

    Article  MathSciNet  MATH  Google Scholar 

  21. J. Dittrich, “Scattering of particles bounded to an infinite planar curve,” arXiv: 1912.03958 [math-ph].

  22. J. Dittrich, P. Exner, Ch. Kühn, and K. Pankrashkin, “On eigenvalue asymptotics for strong \(\delta \)-interactions supported by surfaces with boundaries,” Asymptotic Anal. 97 (1–2), 1–25 (2016).

    MathSciNet  MATH  Google Scholar 

  23. J. Dittrich, P. Exner, and P. Šeba, “Dirac operators with a spherically symmetric \(\delta \)-shell interaction,” J. Math. Phys. 30 (12), 2875–2882 (1989).

    Article  MathSciNet  MATH  Google Scholar 

  24. P. Exner, “Leaky quantum graphs: A review,” in Analysis on Graphs and Its Applications (Am. Math. Soc., Providence, RI, 2008), Proc. Symp. Pure Math. 77, pp. 523–564.

    Article  MathSciNet  MATH  Google Scholar 

  25. P. Exner, “Spectral optimization for singular Schrödinger operators,” Oper. Matrices 14 (3), 705–716 (2020).

    Article  MathSciNet  Google Scholar 

  26. P. Exner and M. Fraas, “On the dense point and absolutely continuous spectrum for Hamiltonians with concentric \(\delta \) shells,” Lett. Math. Phys. 82 (1), 25–37 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  27. P. Exner and T. Ichinose, “Geometrically induced spectrum in curved leaky wires,” J. Phys. A: Math. Gen. 34 (7), 1439–1450 (2001).

    Article  MathSciNet  MATH  Google Scholar 

  28. P. Exner and S. Kondej, “Curvature-induced bound states for a \(\delta \) interaction supported by a curve in \(\mathbb R^3\),” Ann. Henri Poincaré 3 (5), 967–981 (2002).

    Article  MathSciNet  MATH  Google Scholar 

  29. P. Exner and S. Kondej, “Bound states due to a strong \(\delta \) interaction supported by a curved surface,” J. Phys. A: Math. Gen. 36 (2), 443–457 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  30. P. Exner and S. Kondej, “Scattering by local deformations of a straight leaky wire,” J. Phys. A: Math. Gen. 38 (22), 4865–4874 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  31. P. Exner and S. Kondej, “Gap asymptotics in a weakly bent leaky quantum wire,” J. Phys. A: Math. Theor. 48 (49), 495301 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  32. P. Exner and S. Kondej, “Aharonov and Bohm versus Welsh eigenvalues,” Lett. Math. Phys. 108 (9), 2153–2167 (2018).

    Article  MathSciNet  MATH  Google Scholar 

  33. P. Exner, S. Kondej, and V. Lotoreichik, “Asymptotics of the bound state induced by \(\delta \)-interaction supported on a weakly deformed plane,” J. Math. Phys. 59 (1), 013051 (2018).

    Article  MathSciNet  MATH  Google Scholar 

  34. P. Exner and H. Kovařík, Quantum Waveguides (Springer, Cham, 2015).

    Book  MATH  Google Scholar 

  35. P. Exner and K. Němcová, “Leaky quantum graphs: Approximations by point-interaction Hamiltonians,” J. Phys. A: Math. Gen. 36 (40), 10173–10193 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  36. P. Exner and K. Pankrashkin, “Strong coupling asymptotics for a singular Schrödinger operator with an interaction supported by an open arc,” Commun. Partial Diff. Eqns. 39 (2), 193–212 (2014).

    Article  MATH  Google Scholar 

  37. P. Exner and M. Tater, “Spectra of soft ring graphs,” Waves Random Media 14 (1), S47–S60 (2004).

    Article  MathSciNet  MATH  Google Scholar 

  38. G. A. Hagedorn and B. Meller, “Resonances in a box,” J. Math. Phys. 41 (1), 103–117 (2000).

    Article  MathSciNet  MATH  Google Scholar 

  39. S. Kondej and V. Lotoreichik, “Weakly coupled bound state of 2-D Schrödinger operator with potential-measure,” J. Math. Anal. Appl. 420 (2), 1416–1438 (2014).

    Article  MathSciNet  MATH  Google Scholar 

  40. V. Lotoreichik and T. Ourmières-Bonafos, “On the bound states of Schrödinger operators with \(\delta \)-interactions on conical surfaces,” Commun. Partial Diff. Eqns. 41 (6), 999–1028 (2016).

    Article  MATH  Google Scholar 

  41. V. Lotoreichik and T. Ourmières-Bonafos, “A sharp upper bound on the spectral gap for graphene quantum dots,” Math. Phys. Anal. Geom. 22 (2), 13 (2019).

    Article  MathSciNet  MATH  Google Scholar 

  42. A. Mas and F. Pizzichillo, “Klein’s paradox and the relativistic \(\delta \)-shell interaction in \(\mathbb R^3\),” Anal. PDE 11 (3), 705–744 (2018).

    Article  MathSciNet  MATH  Google Scholar 

  43. T. Ourmières-Bonafos and K. Pankrashkin, “Discrete spectrum of interactions concentrated near conical surfaces,” Appl. Anal. 97 (9), 1628–1649 (2018).

    Article  MathSciNet  MATH  Google Scholar 

  44. A. Posilicano, “Boundary triples and Weyl functions for singular perturbations of self-adjoint operators,” Methods Funct. Anal. Topol. 10 (2), 57–63 (2004).

    MathSciNet  MATH  Google Scholar 

  45. K. M. Schmidt, “Critical coupling constants and eigenvalue asymptotics of perturbed periodic Sturm–Liouville operators,” Commun. Math. Phys. 211 (2), 645–685 (2000).

    Article  MathSciNet  Google Scholar 

  46. P. Šeba, “Klein’s paradox and the relativistic point interaction,” Lett. Math. Phys. 18 (1), 77–86 (1989).

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

Thanks are due to the referee for useful remarks.

Funding

The research was supported in part by the European Union within the project CZ.02.1.01/0.0/0.0/16 019/0000778.

Author information

Authors and Affiliations

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

    Pavel Exner

  2. Department of Theoretical Physics, Nuclear Physics Institute, Czech Academy of Sciences, Husinec – Řež, čp. 130, 25068, Řež, Czech Republic

    Pavel Exner

Authors
  1. Pavel Exner
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Pavel Exner.

Additional information

Dedicated to my friend Armen Sergeev on the occasion of his jubilee

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Exner, P. Leaky Quantum Structures. Proc. Steklov Inst. Math. 311, 114–128 (2020). https://doi.org/10.1134/S0081543820060073

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0081543820060073

Keywords

Search

Navigation