Arithmetic Quantum Unique Ergodicity for Symplectic Linear Maps of the Multidimensional Torus

Details Arithmetic Quantum Unique Ergodicity for Symplectic Linear Maps of the Multidimensional Torus Arithmetic Quantum Unique Ergodicity for Symplectic Linear Maps of the Multidimensional Torus

2005-10-23

arxiv.org/abs/math-ph/0510079v5

Ann. of Math. 171 (2), 815--879 (2010)

Physics

Mathematical Physics

68 pages

Scientific paper

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 around 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.

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