Resolvent and scattering matrix at the maximum of the potential

TL;DR

This paper analyzes the microlocal structure of the resolvent and scattering matrix of a semi-classical Schrödinger operator at a potential maximum, revealing their Fourier integral operator nature and applications to spectral and scattering theory.

Contribution

It establishes that the resolvent is a semi-classical Fourier integral operator quantizing Lagrangian submanifolds at a hyperbolic fixed point, with applications to spectral and scattering analysis.

Findings

Resolvent is a semi-classical Fourier integral operator.

Structure of spectral function at critical energy is characterized.

Scattering matrix structure is described at the potential maximum.

Abstract

We study the microlocal structure of the resolvent of the semi-classical Schrodinger operator with short range potential at an energy which is a unique non-degenerate global maximum of the potential. We prove that it is a semi-classical Fourier integral operator quantizing the incoming and outgoing Lagrangian submanifolds associated to the fixed hyperbolic point. We then discuss two applications of this result to describing the structure of the spectral function and the scattering matrix of the Schrodinger operator at the critical energy.