Semiclassical measures for higher dimensional quantum cat maps

Abstract

Consider a quantum cat map M associated to a matrix A ∈ Sp(2n, 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 [DJ18, DJN19] on negatively curved surfaces and [Sch21] 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.

Macroscopic limits of chaotic eigenfunctions

“…We finally discuss the quantum cat map analog of the higher-dimensional Conjecture 4.5, by considering quantum cat maps associated to symplectic integer matrices A ∈ Sp(2n, Z). In this setting Dyatlov-Jézéquel [DJ21] proved Theorem 8. Let µ be a semiclassical measure for a quantum cat map associated to a matrix A ∈ Sp(2n, Z) such that:…”

“…See Figure 1 for a numerical illustration. In particular, we will describe full support statements for weak limits -see Theorem 4 and Theorem 8 -proved in [DJ18,DJN21,DJ21]. The key component is the fractal uncertainty principle first introduced by Dyatlov-Zahl [DZ16] and proved by Bourgain-Dyatlov [BD18].…”

The Spectrum of an Almost Maximally Open Quantized Cat Map

“…We review the pseudodifferential calculus on H N , which also may be found in [3], [8], [19]. Let a ∈ C ∞ (T 2 ).…”

“…Most results from pseudodifferential calculus carry over to H N , see [3], [8] for details. In particular, there is an analogue of Lemma 2.1, given as follows.…”

“…Once adjusted in this manner, such maps enjoy many similar results to continuous scattering problems, and are a subject of current research. In [2], Bouzouina and de Biévre prove a version of quantum ergodicity in the setting of quantum cat maps analogous to Shnirelman [20] and Zelditch's [24] celebrated 1974 result (see also [7]), and more recently in [8], Dyatlov and Jézéquel extend that result to a weak form of quantum unique ergodicity, while the strongest analogous statement of quantum unique ergodicity was disproven by Faure, Nonnenmacher, and De Biévre [11]. Meanwhile, various authors have studied the eigenvalues of such maps, which are analogous to scattering resonances and which obey similar fractal Weyl laws connecting their classical and quantum dynamics.…”