The Marchenko representation of reflectionless Jacobi and Schrödinger operators

Abstract

Abstract. We consider Jacobi matrices and Schrödinger operators that are reflectionless on an interval. We give a systematic development of a certain parametrization of this class, in terms of suitable spectral data, that is due to Marchenko. Then some applications of these ideas are discussed.

Inverse scattering for reflectionless Schrödinger operators with integrable potentials and generalized soliton solutions for the KdV equation

“…In Subsection 4.2, we mention an alternative (and more general) definition of reflectionless potentials in terms of the associated Weyl-Titchmarsh m-functions; the corresponding relation was first derived by Marchenko [19] and then turned into definition by Hur a.o. [12]; see also [25,27] for similar treatments of reflectionless Jacobi matrices.…”

“…We conclude this section by listing some properties of the classical and generalized reflection potentials; cf. [7,9,12,19] and the references therein. Assume that q is a reflectionless potential such that the corresponding Schrödinger operator possesses n negative eigenvalues −κ 2 1 < • • • < −κ 2 n and (right) norming constants m 1 , .…”

“…In addition, paper [20] discusses the corresponding generalized soliton solutions of the KdV equation. Later, Hur, McBride, and Remling [12] took that property of the related mfunctions as defining the notion of reflectionless Schrödinger (or Jacobi) operators. In the Schrödinger operator context, they called a locally integrable potential q (or, more precisely, the Schrödinger operator T q ) reflectionless if the corresponding Weyl-Titchmarsh functions m ± satisfy (1.4) a.e.…”

“…A drawback of the approach of [12,20] is that spectral properties of the Schrödinger operators with potentials in B(−µ 2 ) are not easy to get; they are only implicitly encoded in the measure σ related to the m-functions m ± . On the contrary, the approach of Gesztesy a.o.…”

“…To that end, we used the characterisation of generalized reflectionless potentials q due to Marchenko [20] and Hur a.o. [12] and showed that such a q is uniquely determined by three negative spectra, that of T q and its half-line restrictions T + q and T − q by the Dirichlet condition y(0) = 0. As a by-product, we also described all possible discrete spectra of T q , T + q and T − q for generic reflectionless integrable q.…”