Completeness of the eigenfunctions for the Caudrey–Beals–Coifman system

Abstract

The direct and the inverse scattering problem for the first order linear systems of the type Lψ(x,λ)≡(i(d/dx)+q(x)−λJ)ψ(x,λ)=0, J∈𝔥, q(x)∈𝔤J , which generalizes the Zakharov–Shabat system and the systems studied by Caudrey, Beals, and Coifman (CBC) is analyzed herein. Here J is a regular complex constant element of the Cartan subalgebra 𝔥⊆𝔤 of the simple Lie algebra 𝔤 and the potential q(x) vanishes fast enough for ‖x‖ → ∞ taking values in the image 𝔤J of adJ. The CBC results are generalized and the fundamental analytic solution m(x,λ) for any choice of the irreducible finite-dimensional representation V of 𝔤 is constructed. Four pairwise equivalent minimal sets of scattering data for L, invariant with respect to the choice of the representation of 𝔤, are extracted from the asymptotics of m(x,λ) for x → ±∞. From m(x,λ) the resolvent of L is constructed in the adjoint representation Vad and the completeness relation is proven for the eigenfunctions of L in Vad. It is also proven that the discrete spectrum of L consists of the sets of zeroes of certain spectral invariants D+j(λ) of L.