Arithmetic quantum unique ergodicity for symplectic linear maps of the multidimensional torus


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.


Dubi Kelmer

Department of Mathematics
University of Chicago
5734 University Avenue
Chicago, IL 60637-1514
United States