Skip to main content
SpringerLink
Log in

Point-Like Perturbed Fractional Laplacians Through Shrinking Potentials of Finite Range

  • Published:
Complex Analysis and Operator Theory Aims and scope Submit manuscript

Abstract

We construct the rank-one, singular (point-like) perturbations of the d-dimensional fractional Laplacian in the physically meaningful norm-resolvent limit of fractional Schrödinger operators with regular potentials centred around the perturbation point and shrinking to a delta-like shape. We analyse both possible regimes, the resonance-driven and the resonance-independent limit, depending on the power of the fractional Laplacian and the spatial dimension. To this aim, we also qualify the notion of zero-energy resonance for Schrödinger operators formed by a fractional Laplacian and a regular potential.

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. Albeverio, S., Gesztesy, F., Hø egh-Krohn, R., Holden, H.: Point interactions in two dimensions: basic properties, approximations and applications to solid state physics. J. Reine Angew. Math. 380, 87–107 (1987)

    MathSciNet  MATH  Google Scholar 

  2. Albeverio, S., Gesztesy, F., Høegh-Krohn, R.: The low energy expansion in nonrelativistic scattering theory. Ann. Inst. H. Poincaré Sect. A (N.S.) 37, 1–28 (1982)

    MathSciNet  MATH  Google Scholar 

  3. Albeverio, S., Gesztesy, F., Høegh-Krohn, R., Holden, H.: Solvable Models in Quantum Mechanics. Texts and Monographs in Physics. Springer, New York (1988)

    Book  Google Scholar 

  4. Albeverio, S., Gesztesy, F., Høegh-Krohn, R., Kirsch, W.: On point interactions in one dimension. J. Oper. Theory 12, 101–126 (1984)

    MathSciNet  MATH  Google Scholar 

  5. Albeverio, S., Kurasov, P.: Singular Perturbations of Differential Operators. London Mathematical Society Lecture Note Series, vol. 271. Cambridge University Press, Cambridge (2000). (Solvable Schrödinger type operators)

    Book  Google Scholar 

  6. Capelas de Oliveira, E., Vaz, J.J.: Tunneling in fractional quantum mechanics. J. Phys. A 44, 185303 (2011). 17

    Article  MathSciNet  Google Scholar 

  7. Dell’Antonio, G., Michelangeli, A.: Schrödinger operators on half-line with shrinking potentials at the origin. Asymptot. Anal. 97, 113–138 (2016)

    Article  MathSciNet  Google Scholar 

  8. Gallone, M., Michelangeli, A., Ottolini, A.: Kreĭn-Višik-Birman self-adjoint extension theory revisited. SISSA preprint 25/2017/MATE (2017)

  9. Georgiev, V., Michelangeli, A., Scandone, R.: On fractional powers of singular perturbations of the Laplacian. J. Funct. Anal. 275, 1551–1602 (2018)

    Article  MathSciNet  Google Scholar 

  10. Jarosz, S., Vaz, J.J.: Fractional Schrödinger equation with Riesz-Feller derivative for delta potentials. J. Math. Phys. 57, 123506 (2016). 16

    Article  MathSciNet  Google Scholar 

  11. Kato, T.: Perturbation Theory for Linear Operators. Classics in Mathematics. Springer, Berlin (1995) (Reprint of the 1980 edition)

    Book  Google Scholar 

  12. Klaus, M., Simon, B.: Coupling constant thresholds in nonrelativistic quantum mechanics. I. Short-range two-body case. Ann. Phys. 130, 251–281 (1980)

    Article  MathSciNet  Google Scholar 

  13. Konno, R., Kuroda, S.T.: On the finiteness of perturbed eigenvalues. J. Fac. Sci. Univ. Tokyo Sect. I 13, 55–63 (1966)

    MathSciNet  MATH  Google Scholar 

  14. Laskin, N.: Fractional Quantum Mechanics. World Scientific Publishing Co. Pte. Ltd., Hackensack (2018)

    Book  Google Scholar 

  15. Lenzi, E.K., Ribeiro, H.V., dos Santos, M.A.F., Rossato, R., Mendes, R.S.: Time dependent solutions for a fractional Schrödinger equation with delta potentials. J. Math. Phys. 54, 082107 (2013). 8

    Article  MathSciNet  Google Scholar 

  16. Michelangeli, A., Ottolini, A., Scandone, R.: Fractional powers and singular perturbations of quantum differential Hamiltonians. J. Math. Phys. 59, 072106 (2018). 27

    Article  MathSciNet  Google Scholar 

  17. Muslih, S.I.: Solutions of a particle with fractional \(\delta \)-potential in a fractional dimensional space. Int. J. Theoret. Phys. 49, 2095–2104 (2010)

    Article  MathSciNet  Google Scholar 

  18. Nayga, M.M., Esguerra, J.P.: Green’s functions and energy eigenvalues for delta-perturbed space-fractional quantum systems. J. Math. Phys. 57, 022103 (2016). 7

    Article  MathSciNet  Google Scholar 

  19. Oliveira, E.C.D., Costa, F.S., Vaz, J.J.: The fractional Schrödinger equation for delta potentials. J. Math. Phys. 51, 123517 (2010). 16

    Article  MathSciNet  Google Scholar 

  20. Sacchetti, A.: Stationary solutions of a fractional Laplacian with singular perturbation (2018). arXiv:1801.01694

  21. Sandev, T., Petreska, I., Lenzi, E.K.: Time-dependent Schrödinger-like equation with nonlocal term. J. Math. Phys. 55, 092105 (2014). 10

    Article  MathSciNet  Google Scholar 

  22. Simon, B.: Trace Ideals and Their Applications. Mathematical Surveys and Monographs, vol. 120, 2nd edn. American Mathematical Society, Providence (2005)

    MATH  Google Scholar 

  23. Tare, J.D., Esguerra, J.P.H.: Bound states for multiple Dirac-\(\delta \) wells in space-fractional quantum mechanics. J. Math. Phys. 55, 012106 (2014). 10

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. International School for Advanced Studies – SISSA, Via Bonomea 265, 34136, Trieste, Italy

    Alessandro Michelangeli & Raffaele Scandone

  2. Gran Sasso Science Institute-GSSI, Viale F. Crispi 7, 67100, L’Aquila, Italy

    Raffaele Scandone

Authors
  1. Alessandro Michelangeli
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Raffaele Scandone
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Raffaele Scandone.

Additional information

Communicated by Jussi Behrndt, Fabrizio Colombo, Sergey Naboko.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This article is part of the topical collection “Spectral Theory and Operators in Mathematical Physics” edited by Jussi Behrndt, Fabrizio Colombo and Sergey Naboko.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Michelangeli, A., Scandone, R. Point-Like Perturbed Fractional Laplacians Through Shrinking Potentials of Finite Range. Complex Anal. Oper. Theory 13, 3717–3752 (2019). https://doi.org/10.1007/s11785-019-00927-w

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11785-019-00927-w

Keywords

Mathematics Subject Classification

Search

Navigation