Using basis sets of scar functions

TL;DR

This paper introduces a method for efficiently computing eigenfunctions of chaotic systems by selecting optimal scar functions along periodic orbits, demonstrated on a highly chaotic quartic oscillator.

Contribution

The paper proposes a modified Gram-Schmidt procedure to select the most relevant scar functions, improving eigenfunction computation efficiency for chaotic systems.

Findings

Successfully computed eigenfunctions with a small basis set

Provided estimates of basis size via participation ratio

Analyzed eigenstates using multiple indicators

Abstract

We present a method to efficiently compute the eigenfunctions of classically chaotic systems. The key point is the definition of a modified Gram-Schmidt procedure which selects the most suitable elements from a basis set of scar functions localized along the shortest periodic orbits of the system. In this way, one benefits from the semiclassical dynamical properties of such functions. The performance of the method is assessed by presenting an application to a quartic two dimensional oscillator whose classical dynamics are highly chaotic. We have been able to compute the eigenfunctions of the system using a small basis set. An estimate of the basis size is obtained from the mean participation ratio. A thorough analysis of the results using different indicators, such as eigenstate reconstruction in the local representation, scar intensities, participation ratios, and error bounds, is also presented.