Recent results on quantum map eigenstates

Abstract

One of the central problems in quantum chaos is to obtain a good understanding of the semi-classical behaviour of the eigenfunctions of quantum systems that have a chaotic Hamiltonian system as their classical limit. The pivotal result in this context, and the only general one to date, is the Schnirelman Theorem. Loosely speaking, it states that, if the underlying classical dynamics is ergodic on the appropriate energy surface, then "most" eigenfunctions of the quantum system equidistribute on this energy surface. The challenge is to go beyond this theorem. I will report here on the progress that has been made in this direction in recent years for the special case of quantum maps on the torus. © Springer 2006.