Semiclassical matrix elements for a chaotic propagator in the Scar functions basis

TL;DR

This paper derives a semiclassical approximation for quantum chaotic propagator matrix elements in the scar function basis, expressed using canonical invariants, and verifies it with a linear cat map example.

Contribution

It introduces a new semiclassical formula for matrix elements in the scar basis, utilizing invariant objects and linearization near unstable periodic orbits.

Findings

Derived an explicit semiclassical expression for matrix elements.

Verified the formula's accuracy with a linear cat map.

Applicable to chaotic systems with unstable periodic orbits.

Abstract

A semiclassical approximation for the matrix elements of a quantum chaotic propagator in the scar function basis has been derived. The obtained expression is solely expressed in terms of canonical invariant objects. For our purpose, we have used, the recently developed, semiclassical matrix elements of the propagator in coherent states, together with the linearization of the flux in the neighborhood of a classically unstable periodic orbit of chaotic two dimensional systems. The expression here derived is successfully verified to be exact for a (linear) cat map, after the theory is adapted to a discrete phase space appropriate to a quantized torus.