2021/12/22 by T. J. Christiansen, Christiansen, T. J., A. Uribe +1 · 1 citation
Mathematics · #Spectral Theory in Mathematical Physics #Numerical methods in inverse problems #Mathematical Analysis and Transform Methods
paper · pdf · doi:10.48550/arxiv.2112.12007
We study the microlocal properties of the scattering matrix associated to the semiclassical Schrödinger operator P=h2ΔX+V on a Riemannian manifold with an infinite cylindrical end. The scattering matrix at E=1 is a linear operator S=Sh defined on a Hilbert subspace of L2(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.