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Multichannel scattering for the Schrödinger equation on a line with different thresholds at both infinities

P. O. Kazinski, P. S. Korolev

Abstract

The multichannel scattering problem for the stationary Schrödinger 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.

Multichannel scattering for the Schrödinger equation on a line with different thresholds at both infinities

Abstract

The multichannel scattering problem for the stationary Schrödinger 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 -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.
Paper Structure (10 sections, 6 theorems, 81 equations)

This paper contains 10 sections, 6 theorems, 81 equations.

Table of Contents

  1. Introduction
  2. Jost solutions and their analytical properties
  3. Analytical properties of the transition matrix
  4. Basic identities for the scattering matrix
  5. Identities in the subspace of open channels
  6. Unitarity in the subspace of open channels
  7. Bound states
  8. Shortwave asymptotics
  9. Conclusion
  10. Acknowledgments.

Key Result

Theorem 4.1

If $\lambda\in \mathbb{R}$ belongs to none of the cuts of the functions $(K_\pm)_s$, $s=\overline{1,N}$, i.e., when all the scattering channels are open, the $S$-matrix is unitary

Theorems & Definitions (13)

  • Theorem 4.1
  • proof
  • Remark
  • Theorem 6.1
  • proof
  • Proposition 1
  • Remark
  • proof
  • Proposition 2
  • proof
  • ...and 3 more