2002/11/08 by Andrew Hassell, Steve Zelditch, Hassell, Andrew +1 · 1 citation
Mathematics · #35P99 #58J40 #58J50 #81S10 #Analysis of PDEs (math.AP) #FOS: Mathematics #Spectral Theory (math.SP) #math.AP #math.SP #msc:35P99 #msc:58J40 #msc:58J50 #msc:81S10
paper · pdf · doi:10.48550/arxiv.math/0211140
47 pages, no figures. Only minor changes from the previous version
arxiv created 2003/11/24 · arxiv updated 2009/11/30
Suppose that Omega is a bounded, piecewise smooth Euclidean domain. We prove that the boundary values (Cauchy data) of eigenfunctions of the Laplacian on Omega with various boundary conditions are quantum ergodic if the classical billiard map (on the ball bundle of the boundary of Omega) is ergodic. Our proof is based on the classical observation that the boundary value of an interior eigenfunction is an eigenfunction of an operator Fh on the boundary of Omega, where h-2 is equal to the eigenvalue. In the case of the Neumann boundary condition, Fh is the boundary integral operator induced by the double layer potential. We show that Fh is a semiclassical Fourier integral operator quantizing the billiard map plus a `small' remainder. We then use the quantum dyanmics defined by Fh at the boundary to prove the result, much as the quantum dynamics generated by the wave group were exploited in the work of Zelditch-Zworski (1996). Novelties include the facts that Fh is not unitary and (consequently) the boundary values are equidistributed by measures which are not invariant under the billiard map and which depend on the boundary conditions.