On the Lebesgue Component of Semiclassical Measures for Abelian Quantum Actions

TL;DR

This paper investigates the structure of semiclassical measures for quantum systems associated with symplectic integer matrices, showing that Lebesgue components dominate in weight, especially in irreducible cases, revealing insights into quantum ergodicity.

Contribution

It establishes lower bounds on the Lebesgue component in semiclassical measures for a broad class of symplectic integer matrices, extending understanding of quantum ergodicity on tori.

Findings

Semiclassical measures are convex combinations of Lebesgue and zero entropy measures.

Lebesgue components have weight at least 1/2 in irreducible cases.

In reducible cases, Lebesgue components along invariant subtori sum to at least 1/2.

Abstract

For a large class of symplectic integer matrices, the action on the torus extends to a symplectic $\mathbb{Z}^r$-action with $r\geq 2$. We apply this to the study of semiclassical measures for joint eigenfunctions of the quantization of the symplectic matrices of the $\mathbb{Z}^r$-action. In the irreducible setting, we prove that the resulting probability measures are convex combinations of the Lebesgue measure with weight $\geq 1/2$ and a zero entropy measure. We also provide a general theorem in the reducible case showing that the Lebesgue components along isotropic and symplectic invariant subtori must have total weight $\geq 1/2$.