Geodesic
Arithmetic quantum unique ergodicity for symplectic linear maps of the multidimensional torus
Dubi Kelmer1
1Department of Mathematics<br/>University of Chicago<br/>5734 University Avenue<br/>Chicago, IL 60637-1514<br/>United States
Annals of mathematics, Tome 171 (2010) no. 2, pp. 815-879
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We look at the expectation values for quantized linear symplectic maps on the multidimensional torus and their distribution in the semiclassical limit. We construct super-scars that are stable under the arithmetic symmetries of the system and localize on invariant manifolds. We show that these super-scars exist only when there are isotropic rational subspaces, invariant under the linear map. In the case where there are no such scars, we compute the variance of the fluctuations of the matrix elements for the desymmetrized system and present a conjecture for their limiting distributions.

DOI : 10.4007/annals.2010.171.815

Dubi Kelmer  1

1 Department of Mathematics<br/>University of Chicago<br/>5734 University Avenue<br/>Chicago, IL 60637-1514<br/>United States 
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Dubi Kelmer. Arithmetic quantum unique ergodicity  for symplectic linear maps of the  multidimensional torus. Annals of mathematics, Tome 171 (2010) no. 2, pp. 815-879. doi: 10.4007/annals.2010.171.815

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