The spectral density of the scattering matrix for high energies

TL;DR

This paper derives an explicit formula for the spectral density of the scattering matrix at high energies for Schrödinger operators with short-range potentials, linking it to the X-ray transform of the potential.

Contribution

It provides a novel explicit formula for the eigenvalue density of the scattering matrix in the high energy limit, connecting spectral properties to the potential's X-ray transform.

Findings

Explicit formula for spectral density in high energy regime

Connection between eigenvalue distribution and X-ray transform

Advances understanding of scattering matrix spectral properties

Abstract

We determine the density of eigenvalues of the scattering matrix of the Schrodinger operator with a short range potential in the high energy asymptotic regime. We give an explicit formula for this density in terms of the X-ray transform of the potential.