Using basis sets of scar functions

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.