Semiclassical Measures for Higher-Dimensional Quantum Cat Maps
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23_2023_Article_1309.pdf
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2.5 MB
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Author(s)
Dyatlov, Semyon
Jézéquel, Malo
Date Issued
April 13, 2023
Publisher
Springer International Publishing
Citation
Dyatlov, Semyon and Jézéquel, Malo. 2023. "Semiclassical Measures for Higher-Dimensional Quantum Cat Maps."
Version
Final published version
Abstract
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.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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DOI of Published Version
https://doi.org/10.1007/s00023-023-01309-x
