Multichannel scattering for the Schr\"{o}dinger equation on a line with different thresholds at both infinities

TL;DR

This paper analyzes the multichannel scattering problem for the Schrödinger equation on a line with different thresholds at both infinities, focusing on the structure of solutions, unitarity, and bound states.

Contribution

It provides a comprehensive analysis of the analytical structure of Jost solutions and the transition matrix, including unitarity and symmetry relations, for the Schrödinger equation with asymmetric thresholds.

Findings

Proved unitarity of the scattering matrix with closed channels and different thresholds.

Established symmetry relations of the S-matrix.

Derived asymptotics of Jost functions and transition matrix for large spectral parameters.

Abstract

The multichannel scattering problem for the stationary Schr\"{o}dinger equation on a line with different thresholds at both infinities is investigated. The analytical structure of the Jost solutions and of the transition matrix relating the Jost solutions as functions of the spectral parameter is described. Unitarity of the scattering matrix is proved in the general case when some of the scattering channels can be closed and the thresholds can be different at left and right infinities on the line. The symmetry relations of the $S$-matrix are established. The condition determining the bound states is obtained. The asymptotics of the Jost functions and of the transition matrix are derived for a large spectral parameter.