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.
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Barta, J.: Sur la vibration fundamentale d’une membrane. C. R. Acad. Sci. Paris 204, 472–473 (1937)
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
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
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
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
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
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)
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
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
Frank, R.L.: Eigenvalue bounds for the fractional laplacian: A review. arXiv:1603.09736 (2016)
Georgiev, B., Mukherjee, M.: Nodal geometry, heat diffusion and Brownian motion. Analysis and PDE to appear. arXiv:1602.07110 (2017)
Georgiev, B., Mukherjee, M.: On maximizing the fundamental frequency of the complement of an obstacle. arXiv:1706.02138 (2017)
Georgiev, B., Mukherjee, M., Steinerberger, S.: A spectral gap estimate and applications. arXiv:1612.08565 (2017)
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
Hayman, W.K.: Some bounds for principal frequency. Appl Anal. 7 (3), 247–254 (1977). https://doi.org/10.1080/00036817808839195
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
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
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
Pólya, G., Szegö, G.: Isoperimetric inequalities in mathematical physics annals of mathematics studies, vol. 27. Princeton University Press, Princeton (1951)
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)
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
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
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
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.
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This research of Anup Biswas was supported in part by an INSPIRE faculty fellowship and DST-SERB grant EMR/2016/004810.
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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
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DOI: https://doi.org/10.1007/s11040-017-9256-y
Keywords
- Principal eigenvalue
- Nodal domain
- Fractional Laplacian
- Barta’s inequality
- Ground state
- Fractional Faber-Krahn
- Obstacle problems