Krein systems and canonical systems on a finite interval: accelerants with a jump discontinuity at the origin and continuous potentials

TL;DR

This paper explores the relationship between accelerants and potentials in Krein and canonical systems on finite intervals, showing that continuous potentials can be generated by accelerants with a jump at the origin, which are uniquely determined by the potential.

Contribution

It establishes the existence and uniqueness of accelerants with a jump discontinuity at the origin for continuous potentials in Krein and canonical systems.

Findings

Continuous potentials are generated by accelerants with a jump at the origin.

The generating accelerant is uniquely determined by the potential.

Illustrations provided on pseudo-exponential potentials.

Abstract

This paper is devoted to connections between accelerants and potentials of Krein systems and of canonical systems of Dirac type, both on a finite interval. It is shown that a continuous potential is always generated by an accelerant, provided the latter is continuous with a possible jump discontinuity at the origin. Moreover, the generating accelerant is uniquely determined by the potential. The results are illustrated on pseudo-exponential potentials. The paper is a continuation of the earlier paper of the authors [1] dealing with the direct problem for Krein systems.