Quantum Ergodicity and Mixing of Eigenfunctions

Abstract

This article surveys mathematically rigorous results on quantum ergodic and mixing systems, with an emphasis on general results on asymptotics of eigenfunctions of the Laplacian on compact Riemannian manifolds. Quantum ergodicity and mixing belong to the field of Quantum Chaos, which studies quantizations of ‘chaotic’ classical Hamiltonian systems. The basic questions are, how does the chaos of the classical dynamics impact on the eigenvalues eigenfunctions of the quantum Hamiltonian Ĥ and on and long time dynamics generated by Ĥ? These problems lie at the foundations of the semi-classical limit, i.e. the limit as the Planck constant → 0 or the energy E → ∞. More generally, one could ask what impact any dynamical feature of a classical mechanical system (e.g. complete integrability, KAM, ergodicity) has on the eigenfunctions and eigenvalues of the quantization. Over the last 30 years or so, these questions have been studied rather systematically by both mathematicians and physicists. There is an extensive literature comparing classical and quantum dynamics of model systems, such as comparing the geodesic flow and wave group on a compact (or finite volume) hyperbolic surface, or comparing classical and quantum billiards on the Sinai billiard or the Bunimovich stadium, or comparing the discrete dynamical system generated by a hyperbolic torus automorphism and its quantization by the metaplectic representation. As these models indicate, the basic problems and phenomena are richly embodied in simple, low-dimensional examples in much the same way that two-dimensional toy statistical mechanical models already illustrate complex problems on phase transitions. The principles established for simple models should apply to far more complex systems such as atoms and molecules in strong magnetic fields. The conjectural picture which has emerged from many computer experiments and heuristic arguments on these simple model systems is roughly that there exists a length scale in which quantum chaotic systems exhibit universal behavior. At this length scale, the eigenvalues resemble eigenvalues of random matrices of large size and the eigenfunctions resemble random waves. A small sample of the original physics articles suggesting this picture is [B, BGS, FP, Gu, H, A]. This article reviews some of the rigorous mathematical results in quantum chaos, particularly the rigorous results on eigenfunctions of quantizations of classically ergodic or mixing systems. They support the conjectural picture of random waves up to two moments, i.e. on the level of means and variances. A few results also exist on higher moments in very special cases. But from the mathematical point of view, the conjectural links to random matrices or random waves remain very much open at this time. A key difficulty is that the length scale on which universal behavior should occur is very far below the resolving power of any known Date: January 25, 2005. Research partially supported by NSF grant #DMS 0302518. 1