Stationary scattering theory for $1$-body Stark operators, II

TL;DR

This paper develops a stationary scattering theory for one-body Stark Hamiltonians with short-range potentials, including Coulomb, revealing the scattering matrix as a pseudodifferential operator and analyzing its singularities.

Contribution

It introduces a flexible microlocal analysis approach for Stark operators, extending and improving upon previous methods for understanding scattering matrices.

Findings

Scattering matrix is a classical pseudodifferential operator.

Computed leading order singularities of the scattering matrix kernel.

Extended the analysis to Coulomb and short-range potentials.

Abstract

We study and develop the stationary scattering theory for a class of one-body Stark Hamiltonians with short-range potentials, including the Coulomb potential, continuing our study in [AIIS1,AIIS2]. The classical scattering orbits are parabolas parametrized by asymptotic orthogonal momenta, and the kernel of the (quantum) scattering matrix at a fixed energy is defined in these momenta. We show that the scattering matrix is a classical type pseudodifferential operator and compute the leading order singularities at the diagonal of its kernel. Our approach can be viewed as an adaption of the method of Isozaki-Kitada [IK] used for studying the scattering matrix for one-body Schr\"odinger operators without an external potential. It is more flexible and more informative than the more standard method used previously by Kvitsinsky-Kostrykin [KK1] for computing the leading order singularities of the kernel of the scattering matrix in the case of a constant external field (the Stark case). Our approach relies on Sommerfeld's uniqueness result in Besov spaces, microlocal analysis as well as on classical phase space constructions.