On Quantum Ergodicity for Linear Maps of the Torus

On Quantum Ergodicity for Linear Maps of the Torus

On Quantum Unique Ergodicity for Linear Maps of the Torus

The problem of “quantum ergodicity” addresses the limiting distribution of eigenfunctions of classically chaotic systems. I survey recent progress on this question in the case of quantum maps of the torus. This example leads to analogues of traditional problems in number theory, such as the classical conjecture of Gauss and Artin that any (reasonable) integer is a primitive root for infinitely ...

On Quantum Ergodicity for Linear Maps of the Torus

We prove a strong version of quantum ergodicity for linear hyperbolic maps of the torus (“cat maps”). We show that there is a density one sequence of integers so that as N tends to infinity along this sequence, all eigenfunctions of the quantum propagator at inverse Planck constant N are uniformly distributed. A key step in the argument is to show that for a hyperbolic matrix in the modular gro...

Quantum Mechanics on a Torus *

We present here a canonical description for quantizing classical maps on a torus. We prove theorems analagous to classical theorems on mixing and ergodicity in terms of a quantum Koopman space L (A~, τ~) obtained as the completion of the algebra of observables A~ in the norm induced by the following inner product (A,B) = τ~ ( A†B ) , where τ~ is a linear functional on the algebra analogous to t...

Weyl’s Law and Quantum Ergodicity for Maps with Divided Phase Space

For a general class of unitary quantum maps, whose underlying classical phase space is divided into ergodic and non-ergodic components, we prove analogues of Weyl’s law for the distribution of eigenphases, and the Schnirelman-Zelditch-Colin de Verdière Theorem on the equidistribution of eigenfunctions with respect to the ergodic components of the classical map (quantum ergodicity). We apply our...