Semiclassical Measures for Higher-Dimensional Quantum Cat Maps
Author(s)
Dyatlov, Semyon; Jézéquel, Malo
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Abstract
Consider a quantum cat map M associated with a matrix
$$A\in {{\,\textrm{Sp}\,}}(2n,{\mathbb {Z}})$$
A
∈
Sp
(
2
n
,
Z
)
, which is a common toy model in quantum chaos. We show that the mass of eigenfunctions of M on any nonempty open set in the position–frequency space satisfies a lower bound which is uniform in the semiclassical limit, under two assumptions: (1) there is a unique simple eigenvalue of A of largest absolute value and (2) the characteristic polynomial of A is irreducible over the rationals. This is similar to previous work (Dyatlov and Jin in Acta Math 220(2):297–339, 2018; Dyatlov et al. in J Am Math Soc 35(2):361–465, 2022) on negatively curved surfaces and (Schwartz in The full delocalization of eigenstates for the quantized cat map, 2021) on quantum cat maps with
$$n=1$$
n
=
1
, but this paper gives the first results of this type which apply in any dimension. When condition (2) fails we provide a weaker version of the result and discuss relations to existing counterexamples. We also obtain corresponding statements regarding semiclassical measures and damped quantum cat maps.
Date issued
2023-04-13Department
Massachusetts Institute of Technology. Department of MathematicsPublisher
Springer International Publishing
Citation
Dyatlov, Semyon and Jézéquel, Malo. 2023. "Semiclassical Measures for Higher-Dimensional Quantum Cat Maps."
Version: Final published version