Abstract
We prove that a random orthonormal basis of eigenfunctions on the standard sphere has quantum ergodic behavior.
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[B] Berry, M.V.: Regular and irregular semi-classical wavefunctions. J. Phys. A: Math. Gen.10, 2083–2091 (1977)
[CdeV.1] Colin de Verdière, Y.: Quasi-modes sur les variétés riemannienes compactes. Invent. Math.43, 15–52
[CdeV.2] Colin de Verdière, Y.: Ergodicité et fonctions propres du Laplacien. Sem. Bony-Sjostrand-Meyer, 1984–1985, exposé no. XIII (1985)
[Gu] Guillemin, V.: Band asymptotics in two dimensions. Adv. Math.42, 48–282 (1981)
[G-S] Guillemin, V., Sternberg, S.: Symplectic techniques in physics. Cambridge: Camb. U. Press, 1984
[HMR] Helffer, B., Martinez, A., Robert, D.: Ergodicité et limite semi-classique. Commun. Math. Phys.109, 313–326 (1986)
[I] Ito, K.: Introduction to Probability Theory. Cambridge: Cambridge Univ. Press 1984
[ST] Schrader, R., Taylor, M.: Semi-classical asymptotics, gauge fields, and quantum chaos. J. Funct. Anal.83, 258–316 (1989)
[Sn] Snirelman, A.I.: Ergodic properties of eigenfunctions. Usp. Math. Nank29, 181–182 (1974)
[T] Terras, A.: Harmonic analysis on symmetric spaces and applications 1. Berlin, Heidelberg, New York: Springer 1985
[U] Uribe, A.: A Symbol calculus for a class of Pseudodifferential Operator onS n and band asymptotics. J. Funct. Anal.59, 535–556 (1984)
[V] Voros, A.: Semi-classical ergodicity of quantum eigenstates in the Wigner representation. In: Stochastic Behaviour in Classical and Quantum Hamiltonian Systems. SLNP93, 326–333 (1979)
[Wa] Walters, P.: An introduction to ergodic theory. Berlin, Heidelberg New York: Springer 1982
[Wei] Weinstein, A.: Asymptotics of eingenvalue clusters for the Laplacian plus a potential. Duke Math. J.44, 883–892 (1977)
[Wi] Widom, H.: Eigenvalue distribution theorems for certain homogeneous spaces. J. Funct. Anal.32 (1979)
[Z.1] Zelditch, S.: Quantum transition amplitudes for ergodic and for completely integrable systems. J. Fun. Anal.94, 415–436 (1990)
[Z.2] Zelditch, S.: Uniform distribution of eigenfunctions on compact hyperbolic surfaces. Duke Math. J.55, 919–941 (1987)
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Communicated by Ya.G. Sinai
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Zelditch, S. Quantum ergodicity on the sphere. Commun.Math. Phys. 146, 61–71 (1992). https://doi.org/10.1007/BF02099207
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DOI: https://doi.org/10.1007/BF02099207