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

TL;DR

This paper analyzes the microlocal structure of the scattering matrix for a semiclassical Schrödinger operator on a manifold with an infinite cylindrical end, revealing it as a Fourier integral operator linked to the scattering map.

Contribution

It establishes that, under certain conditions, the scattering matrix is a Fourier integral operator associated with the scattering map, connecting microlocal analysis with geometric scattering theory.

Findings

Scattering matrix is a Fourier integral operator under certain assumptions.

The scattering map is determined by the Hamilton flow of the principal symbol.

Eigenvalues of the scattering matrix are equidistributed on the unit circle under additional hypotheses.

Abstract

We study the microlocal properties of the scattering matrix associated to the semiclassical Schr\"odinger operator $P=h^2\Delta_X+V$ 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 $\kappa:\mathcal{D}_{\kappa}\to T^*Y$, with $\mathcal{D}_\kappa\subset T^*Y$. The scattering map $\kappa$ and its domain $\mathcal{D}_\kappa$ 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.