Weyl Laws for Open Quantum Maps

TL;DR

This paper establishes improved Weyl upper bounds for the eigenvalues of the quantum open baker's map in the semiclassical limit, linking eigenvalue distribution to the trapped set dimension.

Contribution

It provides sharper asymptotic upper bounds for eigenvalues, refining previous results and including explicit dependence on the annulus inner radius for certain cutoff functions.

Findings

Eigenvalue count in an annulus is bounded by O(N^δ)

Improved bounds over previous O(N^{δ+ε}) results

Explicit dependence on annulus inner radius for Gevrey cutoffs

Abstract

We find Weyl upper bounds for the quantum open baker's map in the semiclassical limit. For the number of eigenvalues in an annulus, we derive the asymptotic upper bound $\mathcal O(N^\delta)$ where $\delta$ is the dimension of the trapped set of the baker's map and $(2 \pi N)^{-1}$ is the semiclassical parameter, which improves upon the previous result of $\mathcal O(N^{\delta + \epsilon})$. Furthermore, we derive a Weyl upper bound with explicit dependence on the inner radius of the annulus for quantum open baker's maps with Gevrey cutoffs.