Abstract
Quantum systems whose classical counterpart have ergodic dynamics are quantum ergodic in the sense that almost all eigenstates are uniformly distributed in phase space. In contrast, when the classical dynamics is integrable, there is concentration of eigenfunctions on invariant structures in phase space. In this paper we study eigenfunction statistics for the Laplacian perturbed by a delta-potential (also known as a point scatterer) on a flat torus, a popular model used to study the transition between integrability and chaos in quantum mechanics. The eigenfunctions of this operator consist of eigenfunctions of the Laplacian which vanish at the scatterer, and new, or perturbed, eigenfunctions. We show that almost all of the perturbed eigenfunctions are uniformly distributed in configuration space.
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Bambah R.P., Chowla S.: On numbers which can be expressed as a sum of two squares. Proc. Nat. Inst. Sci. India. 13, 101–103 (1947)
Berkolaiko G., Keating J.P., Winn B.: Intermediate wave functions statistics. Phys. Rev. Lett. 91, 134103 (2003)
Bogomolny, E., Gerland, U., Schmit, C.: Singular statistics. Phys. Rev. E (3) 63(3), part 2, 036206 (2001)
Bogomolny, E., Giraud, O., Schmit C.: Nearest-neighbor distribution for singular billiards. Phys. Rev. E (3) 65(5), 056214 (2002)
Shigehara T., Cheon T.: Wave chaos in quantum billiards with a small but finite-size scatterer. Phys. Rev. E 54, 1321–1331 (1996)
Colin de Verdière Y.: Pseudo-laplaciens. I. Ann. l’Inst.Fourier 32(3), 275–286 (1982)
Colinde Verdière Y.: Ergodicité et fonctions propres du laplacien. Commun. Math. Phys. 102(3), 497–502 (1985)
Hejhal, D.A.: Some observations concerning eigenvalues of the Laplacian and Dirichlet L-series. In: Recent progress in analytic number theory, Symp. Durham 1979, Vol. 2, 1981, pp. 95–110
Huxley, M.N.: Exponential sums and lattice points. III. Proc. London Math. Soc. (3) 87(3), 591–609 (2003)
Jakobson D.: Quantum limits on flat tori. Ann. of Math. (2) 145, 235–266 (1997)
Keating J.P., Marklof J., Winn B.: Localized eigenfunctions in S̆eba billiards. J. Math. Phys. 51(6), 062101 (2010)
Kerckhoff, S., Masurm, H., Smillie, J.: Ergodicity of billiard flows and quadratic differentials. Ann. of Math. (2) 124(2), 293–311 (1986)
de L. Kronig R., Penney W.G.: Quantum Mechanics of Electrons in Crystal Lattices. Proce. Royal Society of London. Series A 130(814), 499–513 (1931)
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)
Marklof J., Rudnick Z.: Almost all eigenfunctions of a rational polygon are uniformly distributed. J. Spectral Th. 2, 107–113 (2012)
Oravecz F., Rudnick Z., Wigman I.: The Leray measure of nodal sets for random eigenfunctions on the torus. Ann. l’Institut Fourier 58(1), 299–335 (2008)
Rahav S., Fishman S.: Spectral statistics of rectangular billiards with localized perturbations. Nonlinearity 15(5), 1541–1594 (2002)
Reed, M., Simon, B.: Methods of Modern Mathematical Physics, Vol. 2: Fourier Analysis and Self- Adjointness. London: Academic Press, 1975
S̆eba P.: Wave Chaos in Singular Quantum Billiard. Phys. Rev. Let. 64(16), 1855–1858 (1990)
Shigehara T.: Conditions for the appearance of wave chaos in quantum singular systems with a pointlike scatterer. Phys. Rev. E 50, 4357–4370 (1994)
Snirel’man, A.: Ergodic properties of eigenfunctions. Usp. Mat. Nauk 29(6), (180), 181–182 (1974)
Corput van der J.G: Neue zahlentheoretische Abschätzungen. Math. Ann. 89(3–4), 215–254 (1923)
Zelditch S.: Uniform distribution of eigenfunctions on compact hyperbolic surfaces. Duke Math. J. 55(4), 919–941 (1987)
Zelditch S., Zworski M.: Ergodicity of eigenfunctions for ergodic billiards. Commun. Math. Phys. 175, 673–682 (1996)
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Rudnick, Z., Ueberschär, H. Statistics of Wave Functions for a Point Scatterer on the Torus. Commun. Math. Phys. 316, 763–782 (2012). https://doi.org/10.1007/s00220-012-1556-2
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DOI: https://doi.org/10.1007/s00220-012-1556-2