On matrix elements for the quantized cat map modulo prime powers

TL;DR

This paper analyzes matrix elements of smooth observables in the quantum cat map modulo prime powers, providing explicit formulas, revealing slower decay of eigenfunction matrix elements, and establishing a limiting distribution for their fluctuations.

Contribution

It introduces explicit exponential sum formulas for matrix elements and demonstrates slower decay and a new limiting distribution in the quantum cat map context.

Findings

Matrix elements decay slower than previously conjectured.

Explicit formulas for matrix elements as exponential sums.

A limiting distribution for the fluctuations of normalized matrix elements.

Abstract

The quantum cat map is a model for a quantum system with underlying chaotic dynamics. In this paper we study the matrix elements of smooth observables in this model, when taking arithmetic symmetries into account. We give explicit formulas for the matrix elements as certain exponential sums. With these formulas we can show that there are sequences of eigenfunctions for which the matrix elements decay significantly slower then was previously conjectured. We also prove a limiting distribution for the fluctuation of the normalized matrix elements around their average.