Resonances for open quantum maps and a fractal uncertainty principle

TL;DR

This paper investigates eigenvalues of quantum open baker's maps with fractal trapped sets, demonstrating a spectral gap larger than standard bounds and establishing a fractal Weyl law for eigenvalue distribution.

Contribution

It introduces a novel spectral gap bound exceeding traditional estimates using a fractal uncertainty principle and derives a fractal Weyl law for eigenvalue counts.

Findings

Spectral gap exceeds standard bound for all fractal dimensions

Spectral gap size can be explicitly computed from the fractal uncertainty principle

Established a fractal Weyl upper bound for eigenvalue counts

Abstract

We study eigenvalues of quantum open baker's maps with trapped sets given by linear arithmetic Cantor sets of dimensions $\delta\in (0,1)$. We show that the size of the spectral gap is strictly greater than the standard bound $\max(0,{1\over 2}-\delta)$ for all values of $\delta$, which is the first result of this kind. The size of the improvement is determined from a fractal uncertainty principle and can be computed for any given Cantor set. We next show a fractal Weyl upper bound for the number of eigenvalues in annuli, with exponent which depends on the inner radius of the annulus.