Affiliations: School of Mathematical Sciences, Tel‐Aviv University, Ramat‐Aviv, 69978 Tel‐Aviv, Israel | Division of Mathematics and Computer Science, Vrije Universiteit, De Boelelaan 1081a, 1081 HV Amsterdam, The Netherlands E‐mail: [email protected] | Division of Mathematics, School of Technology, University of Glamorgan, Pontypridd, CF37 1DL Wales, UK
Note: [] Corresponding author.
Abstract: Scattering problems are solved for a canonical differential system with a pseudo‐exponential potential v. The latter means that v is defined in terms of a triple consisting of an n×n matrix β and two n×m matrices γ1 and γ2 satisfying β*−β=iγ2γ2 *, via the formula v(x)=−2iγ1 *eixα*Σ(x)−1eixαγ, γ=−(γ1+iγ2), where α=β−γ1γ2* and the matrix function Σ is given by Σ(x)=In+∫0xΛ(t)Λ(t)* dt, Λ(x)=[e−ixαγ1 eixαγ]. Such a potential may not be summable. Explicit formulas are presented for the scattering function and the reflection coefficient of the system, and for other functions defined in terms of the asymptotic properties of the fundamental solution of the corresponding differential system. The corresponding inverse problems are also solved explicitly. The state space method from algebraic system theory is used as a basic tool. The results extend those in [11, 2, 5, 4].
