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Scarring for quantum maps with simple spectrum

Part of: Dynamical systems with hyperbolic behavior General mathematical topics and methods in quantum theory

Published online by Cambridge University Press:  18 March 2011

Dubi Kelmer
Dubi Kelmer*
Affiliation:
Department of Mathematics, University of Chicago, 5734 S. University Avenue Chicago, Illinois 60637, USA (email: kelmerdu@math.uchicago.edu)
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Abstract

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In [Kelmer, Scarring on invariant manifolds for perturbed quantized hyperbolic toral automorphisms, Comm. Math. Phys. 276 (2007), 381–395] we introduced a family of symplectic maps of the torus whose quantization exhibits scarring on invariant co-isotropic submanifolds. The purpose of this note is to show that in contrast to other examples, where failure of quantum unique ergodicity is attributed to high multiplicities in the spectrum, for these examples the spectrum is (generically) simple.

Keywords

quantum ergodicityscarringquantum maps

MSC classification

Secondary: 81Q50: Quantum chaos 37D20: Uniformly hyperbolic systems (expanding, Anosov, Axiom A, etc.)
Type
Research Article
Information
Compositio Mathematica , Volume 147 , Issue 5 , September 2011 , pp. 1608 - 1612
Copyright
Copyright © Foundation Compositio Mathematica 2011

References

[AN07]Anantharaman, N. and Nonnenmacher, S., Entropy of semiclassical measures of the Walsh-quantized baker’s map, Ann. Henri Poincaré 8 (2007), 37–74.CrossRefGoogle Scholar
[FND03]Faure, F., Nonnenmacher, S. and De Bièvre, S., Scarred eigenstates for quantum cat maps of minimal periods, Comm. Math. Phys. 239 (2003), 449–492.CrossRefGoogle Scholar
[Kel07]Kelmer, D., Scarring on invariant manifolds for perturbed quantized hyperbolic toral automorphisms, Comm. Math. Phys. 276 (2007), 381–395.CrossRefGoogle Scholar
[Kel10]Kelmer, D., Arithmetic quantum unique ergodicity for symplectic linear maps of the multidimensional torus, Ann. of Math. (2) 171 (2010), 815–879.CrossRefGoogle Scholar
[KR00]Kurlberg, P. and Rudnick, Z., Hecke theory and equidistribution for the quantization of linear maps of the torus, Duke Math. J. 103 (2000), 47–77.CrossRefGoogle Scholar
[Lin06]Lindenstrauss, E., Invariant measures and arithmetic quantum unique ergodicity, Ann. of Math. (2) 163 (2006), 165–219.CrossRefGoogle Scholar
[RS94]Rudnick, Z. and Sarnak, P., The behaviour of eigenstates of arithmetic hyperbolic manifolds, Comm. Math. Phys. 161 (1994), 195–213.CrossRefGoogle Scholar