COMPLETENESS OF THE EIGENFUNCTIONS FOR THE CAUDREY-BEALS-COIFMAN SYSTEM

The direct and the inverse scattering problem for the first order linear systems of the type Lpsi(x, lambda) = (i(d/dx) + q(x) - lambdaJ)psi(x,lambda) = 0, J is-an-element-of h, q(x) is-an-element-of g(j), 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 subset-of g of the simple Lie algebra g and the potential q(x) vanishes fast enough for Absolute value of x --> infinity taking values in the image g(J) of ad(J). The CBC results are generalized and the fundamental analytic solution m(x,lambda) 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, lambda) for x --> +/-infinity. From m(x, lambda) the resolvent of L is constructed in the adjoint representation V(ad) and the completeness relation is proven for the eigenfunctions of L in V(ad). It is also proven that the discrete spectrum of L consists of the sets of zeroes of certain spectral invariants D(j)+(lambda) of L.