Quantum ergodicity of boundary values of eigenfunctions

Suppose that Ω ⊂ ℝⁿ is a bounded, piecewise smooth domain. We prove that the boundary values (Cauchy data) of eigenfunctions of the Laplacian on Ω with various boundary conditions are quantum ergodic if the classical billiard map β on the ball bundle B*(∂Ω) is ergodic. Our proof is based on the classical observation that the boundary values of an interior eigenfunction φ_λ, Δφ_λ = λ²φ_λ is an eigenfunction of an operator F_h on the boundary of Ω with h = λ^-1. In the case of the Neumann boundary condition, F^h is the boundary integral operator induced by the double layer potential. We show that F^h is a semiclassical Fourier integral operator quantizing the billiard map plus a 'small' remainder; the quantum dynamics defined by F_h can be exploited on the boundary much as the quantum dynamics generated by the wave group were exploited in the interior of domains with corners and ergodic billiards in the work of Zelditch-Zworski (1996). Novelties include the facts that F_h is not unitary and (consequently) the boundary values are equidistributed by measures which are not invariant under β and which depend on the boundary conditions. Ergodicity of boundary values of eigenfunctions on domains with ergodic billiards was conjectured by S. Ozawa (1993), and was almost simultaneously proved by Gerard-Leichtnam ( 1993) in the case of convex C^1,1 domains (with continuous tangent planes) and with Dirichlet boundary conditions. Our methods seem to be quite different. Motivation to study piecewise smooth domains comes from the fact that almost all known ergodic domains are of this form.