Semiclassical measures for higher dimensional quantum cat maps

Abstract

Consider a quantum cat map $M$ associated to a matrix $A\in \mathrm{Sp}(2n,\mathbb{Z})$, 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 [arXiv:1705.05019], [arXiv:1906.08923] on negatively curved surfaces and [arXiv:2103.06633] on quantum cat maps with $n=1$, 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.

63 pages, 4 figures. Various revisions following the referee comments. Electronic copy of final peer-reviewed manuscript accepted for publication in Annales Henri Poincaré

References in corpus (2)

• Around quantum ergodicity
• The full delocalization of eigenstates for the quantized cat map

Cited by in corpus (5)

• Uniform observable error bounds of Trotter formulae for the semiclassical Schrödinger equation
• The full delocalization of eigenstates for the quantized cat map
• Semiclassical measures for complex hyperbolic quotients
• Macroscopic limits of chaotic eigenfunctions
• Emergence of quantum dynamics from chaos: The case of prequantum cat maps