Scattering problems for a canonical system with a pseudo‐exponential potential

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^*e^ixα*Σ(x)⁻¹e^ixαγ, γ=−(γ₁+iγ₂), where α=β−γ₁γ₂^* and the matrix function Σ is given by Σ(x)=I_n+∫₀^xΛ(t)Λ(t)^* dt, Λ(x)=[e^−ixαγ₁ e^ixαγ].

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].