Skip to main content
SpringerLink
Log in

Location of Maximizers of Eigenfunctions of Fractional Schrödinger’s Equations

  • Published:
Mathematical Physics, Analysis and Geometry Aims and scope Submit manuscript

Abstract

Eigenfunctions of the fractional Schrödinger operators in a domain D are considered, and a relation between the supremum of the potential and the distance of a maximizer of the eigenfunction from ∂ D is established. This, in particular, extends a recent result of Rachh and Steinerberger arXiv:1608.06604 (2017) to the fractional Schrödinger operators. We also propose a fractional version of the Barta’s inequality and also generalize a celebrated Lieb’s theorem for fractional Schrödinger operators. As applications of above results we obtain a Faber-Krahn inequality for non-local Schrödinger 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

Similar content being viewed by others

References

  1. Barta, J.: Sur la vibration fundamentale d’une membrane. C. R. Acad. Sci. Paris 204, 472–473 (1937)

    MATH  Google Scholar 

  2. Bass, R.F., Levin, D.A.: Harnack inequalities for jump processes. Potential Anal. 17(4), 375–388 (2002). https://doi.org/10.1023/A:1016378210944

    Article  MathSciNet  MATH  Google Scholar 

  3. van den Berg, M.: On the \(L^{\infty }\) norm of the first eigenfunction of the Dirichlet Laplacian. Potential Anal. 13(4), 361–366 (2000). https://doi.org/10.1023/A:1026452623177

    Article  MathSciNet  MATH  Google Scholar 

  4. Bogdan, K., Byczkowski, T., Kulczycki, T., Ryznar, M., Song, R., Vondraček, Z.: Potential analysis of stable processes and its extensions, Lecture Notes in Mathematics, vol. 1980. Springer-Verlag, Berlin (2009). https://doi.org/10.1007/978-3-642-02141-1. Edited by Piotr Graczyk and Andrzej Stos

    Google Scholar 

  5. Carmona, R., Masters, W.C., Simon, B.: Relativistic Schrödinger operators: asymptotic behavior of the eigenfunctions. J. Funct. Anal. 91(1), 117–142 (1990). https://doi.org/10.1016/0022-1236(90)90049-Q

    Article  MathSciNet  MATH  Google Scholar 

  6. Chen, Z.Q., Song, R.: Two-sided eigenvalue estimates for subordinate processes in domains. J. Funct. Anal. 226(1), 90–113 (2005). https://doi.org/10.1016/j.jfa.2005.05.004

    Article  MathSciNet  MATH  Google Scholar 

  7. Chiti, G.: An isoperimetric inequality for the eigenfunctions of linear second order elliptic operators. Boll. Un. Mat. Ital. A (6) 1(1), 145–151 (1982)

    MathSciNet  MATH  Google Scholar 

  8. Chiti, G.: A reverse Hölder inequality for the eigenfunctions of linear second order elliptic operators. Z. Angew. Math. Phys. 33(1), 143–148 (1982). https://doi.org/10.1007/BF00948319

    Article  MathSciNet  MATH  Google Scholar 

  9. De Carli, L., Hudson, S.M.: A Faber-Krahn inequality for solutions of Schrödinger’s equation. Adv. Math. 230(4-6), 2416–2427 (2012). https://doi.org/10.1016/j.aim.2012.04.014

    Article  MathSciNet  MATH  Google Scholar 

  10. Frank, R.L.: Eigenvalue bounds for the fractional laplacian: A review. arXiv:1603.09736 (2016)

  11. Georgiev, B., Mukherjee, M.: Nodal geometry, heat diffusion and Brownian motion. Analysis and PDE to appear. arXiv:1602.07110 (2017)

  12. Georgiev, B., Mukherjee, M.: On maximizing the fundamental frequency of the complement of an obstacle. arXiv:1706.02138 (2017)

  13. Georgiev, B., Mukherjee, M., Steinerberger, S.: A spectral gap estimate and applications. arXiv:1612.08565 (2017)

  14. Harrell II, E.M., Kröger, P., Kurata, K.: On the placement of an obstacle or a well so as to optimize the fundamental eigenvalue. SIAM J. Math. Anal. 33(1), 240–259 (2001). https://doi.org/10.1137/S0036141099357574

    Article  MathSciNet  MATH  Google Scholar 

  15. Hayman, W.K.: Some bounds for principal frequency. Appl Anal. 7 (3), 247–254 (1977). https://doi.org/10.1080/00036817808839195

    Article  MathSciNet  MATH  Google Scholar 

  16. Herbst, I.W.: Spectral theory of the operator (p 2 + m 2)1/2 − Z e 2/r. Comm. Math. Phys. 53(3), 285–294 (1977). http://projecteuclid.org/euclid.cmp/1103900706

    Article  ADS  MathSciNet  Google Scholar 

  17. Lieb, E.H.: On the lowest eigenvalue of the Laplacian for the intersection of two domains. Invent. Math. 74(3), 441–448 (1983). https://doi.org/10.1007/BF01394245

    Article  ADS  MathSciNet  MATH  Google Scholar 

  18. Makai, E.: A lower estimation of the principal frequencies of simply connected membranes. Acta Math. Acad. Sci. Hungar. 16, 319–323 (1965). https://doi.org/10.1007/BF01904840

    Article  MathSciNet  MATH  Google Scholar 

  19. Pólya, G., Szegö, G.: Isoperimetric inequalities in mathematical physics annals of mathematics studies, vol. 27. Princeton University Press, Princeton (1951)

    MATH  Google Scholar 

  20. Rachh, M., Steinerberger, S.: On the location of maxima of solutions of schrödinger’s equation. Communications Pure and Applied Mathematics to appear. arXiv:1608.06604 (2017)

  21. Ros-Oton, X., Serra, J.: The Dirichlet problem for the fractional Laplacian: regularity up to the boundary. J. Math. Pures Appl. (9) 101(3), 275–302 (2014). https://doi.org/10.1016/j.matpur.2013.06.003

    Article  MathSciNet  MATH  Google Scholar 

  22. Sato, S.: An inequality for the spectral radius of Markov processes. Kodai Math. J. 8(1), 5–13 (1985). https://doi.org/10.2996/kmj/1138036992

    Article  MathSciNet  MATH  Google Scholar 

  23. Steinerberger, S.: Lower bounds on nodal sets of eigenfunctions via the heat flow. Comm. Partial Diff. Equ. 39(12), 2240–2261 (2014). https://doi.org/10.1080/03605302.2014.942739

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

Special thanks to my colleague Tejas Kalelkar for teaching me Inkscape which has been used to draw the diagrams of this article. The author is indebted to Stefan Steinerberger for constructive comments and suggestions.

Author information

Authors and Affiliations

  1. Department of Mathematics, Indian Institute of Science Education and Research, Dr. Homi Bhabha Road, Pune, 411008, India

    Anup Biswas

Authors
  1. Anup Biswas
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Anup Biswas.

Additional information

This research of Anup Biswas was supported in part by an INSPIRE faculty fellowship and DST-SERB grant EMR/2016/004810.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Biswas, A. Location of Maximizers of Eigenfunctions of Fractional Schrödinger’s Equations. Math Phys Anal Geom 20, 25 (2017). https://doi.org/10.1007/s11040-017-9256-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11040-017-9256-y

Keywords

Mathematics Subject Classification (2010)

Search

Navigation