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Quantum ergodicity of boundary values of eigenfunctions. (English) Zbl 1054.58022

Let \(\Omega\) be a bounded, piecewise smooth domain in \({R}^n\). Assume that the classical billiard map on the ball bundle \(B^*(\partial\Omega)\) is ergodic. Impose Dirichlet, Neumann, Robin boundary conditions, or \(\Psi^1\) Robin boundary conditions.
The authors show that the boundary values (Cauchy data) of the eigenfunctions of the Laplacian on \(\Omega\) are quantum ergodic.
The paper is organized into sections: 1) Introduction, 2) Quantum ergodicity of endomorphisms, 3) Piecewise smooth manifolds, 4) Structure of the operators \(E_h\) and \(F_h\), 5) Local Weyl law, 6) Egorov theorem, 7) Proof of main theorem - Neumann boundary conditions, 8) Robin boundary conditions, 9) Dirichlet boundary conditions, 10) \(\Psi^1\)-Robin boundary conditions, 11) non-convex domains, 12) Appendix - heat kernel.

MSC:

58J50 Spectral problems; spectral geometry; scattering theory on manifolds
81S10 Geometry and quantization, symplectic methods
58J40 Pseudodifferential and Fourier integral operators on manifolds
46L60 Applications of selfadjoint operator algebras to physics
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[1] Bunimovich, L.A., Rehacek, J.: On the ergodicity of many-dimensional focusing billiards. Classical and quantum chaos. Ann. Inst. H. Poincaré Phys. Théor. 68(4), 421–448 (1998) · Zbl 0907.58037
[2] Bunimovich, L.A., Rehacek, J.: How high-dimensional stadia look like. Commun. Math. Phys. 197(2), 277–301 (1998) · Zbl 1022.37021 · doi:10.1007/s002200050451
[3] Bunimovich, L.A.: Private communication
[4] Bunimovich, L.A.: Conditions of stochasticity of two-dimensional billiards. Chaos 1(2), 187–193 (1991) · Zbl 0899.58039 · doi:10.1063/1.165827
[5] Bunimovich, L.A.: On the ergodic properties of nowhere dispersing billiards. Commun. Math. Phys. 65(3), 295–312 (1979) · Zbl 0421.58017 · doi:10.1007/BF01197884
[6] Burq, N.: Quantum ergodicity of boundary values of eigenfunctions: A control theory approach. arXiv:math.AP/0301349, 2003
[7] Colin de Verdière, Y.: Ergodicité et fonctions propres du laplacien. Commun. Math. Phys. 102, 497–502 (1985) · Zbl 0592.58050 · doi:10.1007/BF01209296
[8] Cornfeld, I.P., Fomin, S.V., Sinai, Ya.G.: Ergodic theory. Grundelehren Math. Wiss. 245, Berlin: Springer, 1982 · Zbl 0493.28007
[9] Davies, E.B.: Heat kernels and spectral theory. Cambridge Tracts in Mathematics 92, Cambridge: Cambridge University Press, 1989 · Zbl 0699.35006
[10] Dimassi, M., Sjöstrand, J.: Spectral Asymptotics in the Semi-classical Limit. London Mathematical Society Lecture Note Series 268, Cambridge: Cambridge University Press, 1999 · Zbl 0926.35002
[11] Gerard, P., Leichtnam, E.: Ergodic properties of eigenfunctions for the Dirichlet problem. Duke Math. J. 71, 559–607 (1993) · Zbl 0788.35103 · doi:10.1215/S0012-7094-93-07122-0
[12] Hassell, A., Tao, T.: Upper and lower bounds for normal derivatives of Dirichlet eigenfunctions. Math. Res. Lett. 9, 289–307 (2002) · Zbl 1014.58015 · doi:10.4310/MRL.2002.v9.n3.a6
[13] Helffer, B., Sjöstrand, J.: Équation de Schrödinger avec champ magnétique et équation de Harper. Lecture Notes in Physics 345, Berlin-Heidelberg-New York: Springer, 1989, pp. 118–197 · Zbl 0699.35189
[14] Hörmander, L.: The analysis of linear partial differential operators, Vol. 1, second edition, Berlin: Springer-Verlag, 1990 · Zbl 0687.35002
[15] Hörmander, L.: The analysis of linear partial differential operators, Vol. 4, Berlin: Springer-Verlag, 1985 · Zbl 0601.35001
[16] Kerckhoff, S., Masur, H., Smillie, J.: Ergodicity of billiard flows and quadratic differentials. Ann. of Math. (2) 124(2), 293–311 (1986) · Zbl 0637.58010
[17] Li, C., McIntosh, A., Semmes, S.: Convolution integrals on Lipschitz surfaces. J. Am. Math. Soc. 5, 455–481 (1992) · Zbl 0763.42009 · doi:10.1090/S0894-0347-1992-1157291-5
[18] Ozawa, S.: Asymptotic property of eigenfunction of the Laplacian at the boundary. Osaka J. Math. 30, 303–314 (1993) · Zbl 0808.35090
[19] Ozawa, S.: Hadamard’s variation of the Green kernels of heat equations and their traces. I. J. Math. Soc. Japan 34(3), 455–473 (1982) · Zbl 0476.35039 · doi:10.2969/jmsj/03430455
[20] Ozawa, S.: Peturbation of domains and Green kernels of heat equations. Proc. Japan. Acad. Soc. Japan 54, 322–325 (1978) · Zbl 0432.35003 · doi:10.3792/pjaa.54.322
[21] Ozawa, S.: The eigenvalues of the Laplacian and perturbation of boundary conditions, Proc. Japan. Acad. Soc. Japan 55, 121–3 (1979) · Zbl 0434.35069
[22] Paul, T., Uribe, A.: The semi-classical trace formula and propagation of wave packets. J. Funct. Anal. 132, 192–249 (1995) · Zbl 0837.35106 · doi:10.1006/jfan.1995.1105
[23] Petersen, K.: Ergodic Theory, Cambridge studies in advanced mathematics 2, Cambridge: Cambridge University Press, 1983
[24] Schnirelman, A.I.: Ergodic properties of eigenfunctions. Usp. Math. Nauk. 29, 181–2 (1974)
[25] Seeley, R.: The resolvent of an elliptic boundary problem. Am. J. Math. 91, 889–920 (1969) · Zbl 0191.11801 · doi:10.2307/2373309
[26] Seeley, R.: Analytic extension of the trace associated with elliptic boundary problems. Am. J. Math. 91, 963–983 (1969) · Zbl 0191.11901 · doi:10.2307/2373312
[27] Taylor, M.E.: Pseudodifferential Operators. Princeton, NJ: Princeton Mathematical Series, 1981 · Zbl 0453.47026
[28] Taylor, M.E.: Partial differential equations I — Basic theory. Texts in Applied Mathematics 23, New York: Springer-Verlag, 1996 · Zbl 0869.35001
[29] Taylor, M.E.: Partial differential equations. II. Applied Mathematical Sciences 116, New York: Springer-Verlag, 1996 · Zbl 0869.35001
[30] Verchota, G.: Layer potentials and regularity for the Dirichlet problem for Laplace’s equation in Lipschitz domains. J. Funct. Anal. 59, 572–611 (1984) · Zbl 0589.31005 · doi:10.1016/0022-1236(84)90066-1
[31] Wojtkowski, M.: Principles for the design of billiards with nonvanishing Lyapunov exponents. Commun. Math. Phys. 105(3), 391–414 (1986) · Zbl 0602.58029 · doi:10.1007/BF01205934
[32] Zelditch, S.: Uniform distribution of eigenfunctions on compact hyperbolic surfaces. Duke Math. J. 55, 919–941 (1987) · Zbl 0643.58029 · doi:10.1215/S0012-7094-87-05546-3
[33] Zelditch, S.: Quantum ergodicity of C * dynamical systems. Commun. Math. Phys. 177, 507–528 (1996) · Zbl 0856.58019 · doi:10.1007/BF02101904
[34] Zelditch, S.: The inverse spectral problem for analytic plane domains, I: Balian-Bloch trace formula. arXiv: math.SP/0111077 to appear, Commun. Math. Phys. · Zbl 1086.58016
[35] Zelditch, S., Zworski, M.: Ergodicity of eigenfunctions for ergodic billiards. Commun. Math. Phys. 175, 673–682 (1996) · Zbl 0840.58048 · doi:10.1007/BF02099513
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