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On the Maximal Scarring for Quantum Cat Map Eigenstates

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Abstract

We consider the quantized hyperbolic automorphisms on the 2-dimensional torus (or generalized quantum cat maps), and study the localization properties of their eigenstates in phase space, in the semiclassical limit. We prove that if the semiclassical measure corresponding to a sequence of normalized eigenstates has a pure point component (phenomenon of ‘‘strong scarring’’), then the weight of this component cannot be larger than the weight of the Lebesgue component, and therefore admits the sharp upper bound 1/2.

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References

  1. Bonechi, F., De Bièvre, S.: Exponential mixing and ln ℏ timescales in quantized hyperbolic maps on the torus. Commun. Math. Phys. 211, 659–686 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bonechi, F., De Bièvre, S.:Controlling strong scarring for quantized ergodic toral automorphisms. Duke Math J. 117, 571–587 (2003); Section 5 appears only in the preprint mp-arc/02-81 (2002)

    MATH  Google Scholar 

  3. Bouzouina, A., De Bièvre, S.: Equipartition of the eigenfunctions of quantized ergodic maps on the torus. Commun. Math. Phys. 178, 83–105 (1996)

    MathSciNet  MATH  Google Scholar 

  4. Chang, C.-H., Krüger, T., Schubert, R.: Quantizations of piecewise affine maps on the torus and their quantum limits. In preparation

  5. Colin de Verdière, Y.: Ergodicité et fonctions propres du Laplacien. Commun. Math. Phys. 102, 497–502 (1985)

    MATH  Google Scholar 

  6. Corvaja, P., Rudnick, Z., Zannier, U.: A lower bound for periods of matrices. Preprint math. NT/0309215 (2003)

  7. Degli Esposti, M.: Quantization of the orientation preserving automorphisms of the torus. Ann. Inst. Henri Poincaré 58, 323–341 (1993)

    MATH  Google Scholar 

  8. Faure, F., Nonnenmacher, S., De Bièvre, S.: Scarred eigenstates for quantum cat maps of minimal periods, Commun. Math. Phys. 239, 449–492 (2003)

    Google Scholar 

  9. Hannay, J.H., Berry, M.V.: Quantization of linear maps-Fresnel diffraction by a periodic grating. Physica D 1, 267–290 (1980)

    Article  MathSciNet  Google Scholar 

  10. Heller, E.J.: Bound-state eigenfunctions of classically chaotic Hamiltonian systems: Scars of periodic orbits. Phys. Rev. Lett. 53, 1515–1518 (1984)

    Google Scholar 

  11. Koyama, S.: Quantum ergodicity of Eisenstein series for arithmetic 3-manifolds. Commun. Math. Phys. 215, 477–486 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  12. Kurlberg, P., Rudnick, Z.: Hecke theory and equidistribution for the quantization of linear maps of the torus. Duke Math. J. 103, 47–77 (2000)

    MathSciNet  MATH  Google Scholar 

  13. Lindenstrauss, E.: Invariant measures and arithmetic quantum unique ergodicity. Preprint (2003)

  14. Luo, W., Sarnak, P.: Quantum ergodicity of eigenfunctions on PSL 2(ℤ) H 2. Publ. Math. IHES 81, 207–237 (1995)

    MathSciNet  MATH  Google Scholar 

  15. Marcus, B.: A note on periodic points of toral automorphisms. Monatsh. Math. 89, 121–129 (1980)

    MathSciNet  MATH  Google Scholar 

  16. Marklof, J., Rudnick, Z.: Quantum unique ergodicity for parabolic maps. Geom. Funct. Anal. 10, 1554–1578 (2000)

    MathSciNet  MATH  Google Scholar 

  17. Petridis, Y., Sarnak, P.: Quantum unique ergodicity for SL 2(ℴ)\H 3 and estimates for L-functions. J. Evol. Equ. 1, 277–290 (2001)

    MathSciNet  MATH  Google Scholar 

  18. Rudnick, Z., Sarnak, P.: The behaviour of eigenstates of arithmetic hyperbolic manifolds. Commun. Math. Phys. 161, 195–231 (1994)

    MathSciNet  MATH  Google Scholar 

  19. Schnirelman, A.: Ergodic properties of eigenfunctions. Usp. Math. Nauk 29, 181–182 (1974)

    Google Scholar 

  20. Sigmund, K.: Generic Properties of Invariant Measures for Axiom A-Diffeormorphisms. Invent. Math. 11, 99–109 (1970)

    MATH  Google Scholar 

  21. Wolpert, S. A.: The modulus of continuity for Γ0(m)\ℍ semi-classical limits. Commun. Math. Phys. 216, 313–323 (2001)

    MathSciNet  MATH  Google Scholar 

  22. Zelditch, S.: Uniform distribution of the eigenfunctions on compact hyperbolic surfaces. Duke Math. J 55, 919–941 (1987); Index and dynamics of quantized contact transformations, Ann. Inst. Fourier 47, 673–682 (1996)

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Laboratoire de Physique et Modélisation des Milieux Condensés (LPM2C), 66, 38042, Grenoble Cédex 9, France

    Frédéric Faure

  2. Service de Physique Théorique, CEA/DSM/PhT Unité de Recherche Associée au CNRS, CEA/Saclay, 91191, Gif-sur-Yvette, Cédex, France

    Stéphane Nonnenmacher

  3. Mathematical Sciences Research Institute, 1000, Centennial Drive, Berkeley, CA 94720-5070, USA

    Stéphane Nonnenmacher

Authors
  1. Frédéric Faure
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  2. Stéphane Nonnenmacher
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Correspondence to Stéphane Nonnenmacher.

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Communicated by P. Sarnak

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Faure, F., Nonnenmacher, S. On the Maximal Scarring for Quantum Cat Map Eigenstates. Commun. Math. Phys. 245, 201–214 (2004). https://doi.org/10.1007/s00220-003-1019-x

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  • DOI: https://doi.org/10.1007/s00220-003-1019-x

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