Exact Eigenfunctions of a Chaotic System

TL;DR

This paper derives exact eigenfunctions and Green's functions for a class of chaotic quantum systems, tiling billiards on the pseudo-sphere, and compares these results with numerical data, finding no evidence of quantum scars.

Contribution

It provides an exact analytical expression for eigenfunctions and Green's functions in a chaotic system where classical-quantum correspondence is exact, extending understanding beyond semiclassical approximations.

Findings

Exact Green's function expressed as sums over periodic orbits

Eigenfunctions identified from Green's function expressions

No evidence of quantum scars found in the system

Abstract

The interest in the properties of quantum systems, whose classical dynamics are chaotic, derives from their abundance in nature. The spectrum of such systems can be related, in the semiclassical approximation (SCA), to the unstable classical periodic orbits, through Gutzwiller's trace formula. The class of systems studied in this work, tiling billiards on the pseudo-sphere, is special in this correspondence being exact, via Selberg's trace formula. In this work, an exact expression for Green's function (GF) and the eigenfunctions (EF) of tiling billiards on the pseudo-sphere, whose classical dynamics are chaotic, is derived. GF is shown to be equal to the quotient of two infinite sums over periodic orbits, where the denominator is the spectral determinant. Such a result is known to be true for typical chaotic systems, in the leading SCA. From the exact expression for GF, individual EF can be identified. In order to obtain a SCA by finite series for the infinite sums encountered, resummation by analytic continuation in $\hbar$ was performed. The result is similar to known results for EF of typical chaotic systems. The lowest EF of the Hamiltonian were calculated with the help of the resulting formulae, and compared with exact numerical results. A search for scars with the help of analytical and numerical methods failed to find evidence for their existence.