Abstract
We study the small scale distribution of the eigenfunctions of a point scatterer (the Laplacian perturbed by a delta potential) on two- and three-dimensional flat tori. In two dimensions, we establish small scale equidistribution for the “new” eigenfunctions holding all the way down to the Planck scale. In three dimensions, small scale equidistribution is established for all of the “new” eigenfunctions at certain scales.
Similar content being viewed by others
Notes
The normalization by \(2\pi \) is introduced to facilitate the notation below.
References
Berry, M.V.: Regular and irregular semiclassical wavefunctions. J. Phys. A 10(12), 2083–2091 (1977)
Berry, M.V.: Semiclassical mechanics of regular and irregular motion. In: Chaotic Behavior of Deterministic Systems (Les Houches, 1981), pp. 171–271. North-Holland, Amsterdam (1983)
Bourgain, J., Rudnick, Z.: On the geometry of the nodal lines of eigenfunctions of the two-dimensional torus. Ann. Henri Poincaré 12(6), 1027–1053 (2011)
Colin de Verdière, Y.: Pseudo-Laplaciens. I. Ann. Inst. Fourier (Grenoble) 32(3), xiii, 275–286 (1982)
Colin de Verdière, Y.: Ergodicité et fonctions propres du laplacien. Commun. Math. Phys. 102(3), 497–502 (1985)
de Courcy-Ireland, M.: Small-scale equidistribution for random spherical harmonics. Preprint. Available online. arXiv:1711.01317
Granville, A., Wigman, I.: Planck-scale mass equidistribution of toral Laplace eigenfunctions. Commun. Math. Phys. 355(2), 767–802 (2017)
Han, X.: Small scale quantum ergodicity in negatively curved manifolds. Nonlinearity 28(9), 3263–3288 (2015)
Han, X.: Small scale equidistribution of random eigenbases. Commun. Math. Phys. 349(1), 425–440 (2017)
Han, X., Tacy, M.: Equidistribution of random waves on small balls. Preprint. Available online. arXiv:1611.05983
Harman, G.: On the Erdős–Turán inequality for balls. Acta Arith. 85(4), 389–396 (1998)
Hezari, H., Rivière, G.: \(L^{p}\) norms, nodal sets, and quantum ergodicity. Adv. Math. 290, 938–966 (2016)
Hezari, H., Rivière, G.: Quantitative equidistribution properties of toral eigenfunctions. J. Spectr. Theory 7, 471–485 (2017)
Humphries, P.: Equidistribution in shrinking sets and \(L^{4}\)-norm bounds for automorphic forms. Math. Ann. 371, 1497–1543 (2018)
Kurlberg, P., Rosenzweig, L.: Superscars for arithmetic toral point scatterers. Commun. Math. Phys. 349(1), 329–360 (2017)
Kurlberg, P., Ueberschär, H.: Quantum ergodicity for point scatterers on arithmetic tori. Geom. Funct. Anal. 24(5), 1565–1590 (2014)
Kurlberg, P., Ueberschär, H.: Superscars in the Šeba billiard. J. Eur. Math. Soc. (JEMS) 19(10), 2947–2964 (2017)
Landau, E.: Über die Einteilung der positiven ganzen Zahlen in vier Klassen nach der Mindeszahl der zu ihrer additiven Zusammensetzung erforderlichen Quadrate. Arch. Math. Phys. 13, 305–312 (1908)
Lester, S., Rudnick, Z.: Small scale equidistribution of eigenfunctions on the torus. Commun. Math. Phys. 350(1), 279–300 (2017)
Luo, W.Z., Sarnak, P.: Quantum ergodicity of eigenfunctions on \(\text{ PSL }_{2}({\mathbb{Z}})\backslash {\mathbb{H}}^{2}\). Inst. Hautes Études Sci. Publ. Math. 81, 207–237 (1995)
Rudnick, Z., Ueberschär, H.: Statistics of wave functions for a point scatterer on the torus. Commun. Math. Phys. 316(3), 763–782 (2012)
Šeba, P.: Wave chaos in singular quantum billiard. Phys. Rev. Lett. 64(16), 1855–1858 (1990)
Siegel, C.L.: Über die Classenzahl quadratischer Zahlkörper. Acta Arith. 1, 83–86 (1935)
Shigehara, T.: Conditions for the appearance of wave chaos in quantum singular systems with a pointlike scatterer. Phys. Rev. E 50, 4357–4370 (1994)
Shnirel’man, A.I.: Ergodic properties of eigenfunctions. Uspehi Mat. Nauk 29(6(180)), 181–182 (1974)
Ueberschär, H.: Quantum chaos for point scatterers on flat tori. Philos. Trans. R. Soc. Lond. Ser. A. 372, 20120509 (2014)
Wigman, I., Yesha, N.: Central limit theorem for Planck scale mass distribution of toral Laplace eigenfunctions. Mathematika 65(3), 643–676 (2019)
Yesha, N.: Eigenfunction statistics for a point scatterer on a three-dimensional torus. Ann. Henri Poincaré 14(7), 1801–1836 (2013)
Yesha, N.: Quantum ergodicity for a point scatterer on the three-dimensional torus. Ann. Henri Poincaré 16(1), 1–14 (2015)
Yesha, N.: Uniform distribution of eigenstates on a torus with two point scatterers. J. Spectr. Theory 8, 1509–1527 (2018)
Young, M.: The quantum unique ergodicity conjecture for thin sets. Adv. Math. 286, 958–1016 (2016)
Zelditch, S.: Uniform distribution of eigenfunctions on compact hyperbolic surfaces. Duke Math. J. 55(4), 919–941 (1987)
Acknowledgements
The author would like to express his gratitude to Z. Rudnick and I. Wigman for useful discussions and comments. The research leading to these results was partially supported by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013), ERC Grant Agreement No. 335141.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by J. Marklof
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Yesha, N. Small Scale Equidistribution for a Point Scatterer on the Torus. Commun. Math. Phys. 377, 199–224 (2020). https://doi.org/10.1007/s00220-019-03669-0
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00220-019-03669-0