Direct and inverse scattering problem for canonical systems with a strictly pseudo-exponential potential

Explicit formulas are given for the solutions of the direct and inverse scattering problems for a canonical differential system with a strictly pseudo-exponential potential. The proofs are self-contained and employ state space techniques from mathematical system theory. The paper supplements an earlier paper of the first two authors, where explicit formulas were given using Marchenko's approach, and an earlier paper of the last three authors, where self-contained proofs were given for the corresponding direct and inverse spectral problems. Two types of factorizations of the scattering matrix function appear and connections between them are considered.