The Spectrum of an Almost Maximally Open Quantized Cat Map

TL;DR

This paper analyzes eigenvalues of a quantized cat map with phase space restrictions, deriving formulas and bounds in the semiclassical limit, supported by numerical evidence.

Contribution

It provides explicit formulas for eigenvalues of the quantized cat map with phase space cutoffs and establishes decay bounds when fixed points are absent.

Findings

Explicit eigenvalue formulas for the quantized cat map with phase space cutoff.

Superpolynomial decay bounds on eigenvalues without fixed points.

Numerical illustrations confirming theoretical results.

Abstract

We consider eigenvalues of a quantized cat map (i.e. hyperbolic symplectic integer matrix), cut off in phase space to include a fixed point as its only periodic orbit on the torus. We prove a simple formula for the eigenvalues on both the quantized real line and the quantized torus in the semiclassical limit as $h\to0$. We then consider the case with no fixed points, and prove a superpolynomial decay bound on the eigenvalues. The results are illustrated with numerical calculations.