Lecture Notes on Quantum Chaos

We give an overview of the quantum ergodicity result. 1. Quantum ergodicity in the physical space 1.1. Concentration of eigenfunctions. First, let us consider the case when M ⊂ R is a bounded domain with piecewise C∞ boundary and we take the operator ∆ = −∂ x1 − ∂ 2 x2 . We study the Dirichlet eigenvalues λ0 < λ1 ≤ λ2 ≤ . . . (with multiplicities taken into account) and the corresponding L normalized eigenfunctions uj ∈ H 0 (M), so ∆uj = λ 2 juj, uj|∂M = 0, ‖uj‖L2(M) = 1. (1.1) We are interested in the following question regarding the high energy limit: Question 1.1. How do uj concentrate as j →∞? In general, uj become rapidly oscillating at high energies, so we have to study their concentration in some rough sense. A natural way to do that is to take the weak limits of the measures |uj(x)| dx along subsequences: Definition 1.2. Let ujk be a subsequence of (uj) and μ a probability measure on M . We say that ujk → μ weakly, if ∫ M a(x)|ujk(x)|2 dx→ ∫ M a(x) dμ(x) for all a ∈ C∞ 0 (M) (1.2) We say that ujk equidistributes in M , if it converges weakly to the volume measure: ujk → dx Vol(M) . (1.3) Remarks. 1. By a standard density argument, once (1.2) holds for all a ∈ C∞ 0 (M), it holds for all a ∈ C(M). 2. By a diagonal argument (see [Zw, Theorem 5.2]), there always exists a subsequence of uj converging to some measure. 1