The semiclassical structure of the scattering matrix for a manifold with infinite cylindrical end

Abstract

We study the microlocal properties of the scattering matrix associated to the semiclassical Schrödinger operator P = h 2 ∆ X + V on a Riemannian manifold with an infinite cylindrical end. The scattering matrix at E = 1 is a linear operator S = S h defined on a Hilbert subspace of L 2 (Y ) that parameterizes the continuous spectrum of P at energy 1. Here Y is the cross section of the end of X, which is not necessarily connected. We show that, under certain assumptions, microlocally S is a Fourier integral operator associated to the graph of the scattering map κ : Dκ → T * Y , with Dκ ⊂ T * Y . The scattering map κ and its domain Dκ are determined by the Hamilton flow of the principal symbol of P . As an application we prove that, under additional hypotheses on the scattering map, the eigenvalues of the associated unitary scattering matrix are equidistributed on the unit circle.