[Paper Review] Faddeev-Marchenko scattering for CMV matrices and the Strong Szego Theorem

Leonid Golinskiĭ, Alexander Kheifets+2 more|ArXiv.org|Jul 25, 2008

Spectral Theory in Mathematical Physics18 references3 citations

TL;DR

This paper solves the inverse scattering problem for CMV matrices in the Szegö class by establishing necessary and sufficient conditions under which the scattering function uniquely determines the matrix. It leverages three classical theorems—Adamyan-Arov-Kreín (AAK), Helson-Szegö, and Strong Szegö Limit—to characterize subclasses of CMV matrices where the scattering correspondence is one-to-one, providing an explicit reconstruction framework via generalized Gelfand-Levitan-Marchenko operators.

ABSTRACT

B. Simon proved the existence of the wave operators for the CMV matrices with Szego class Verblunsky coefficients, and therefore the existence of the scattering function. Generally, there is no hope to restore a CMV matrix when we start from the scattering function, in particular, because it does not contain any information about the (possible) singular measure. Our main point of interest is the solution of the inverse scattering problem (the heart of the Faddeev--Marchenko theory), that is, to give necessary and sufficient conditions on a certain class of CMV matrices such that the restriction of this correspondence (from a matrix to the scattering function) is one to one. In this paper we show that the main questions on inverse scattering can be solved with the help of three important classical results: Adamyan-Arov-Krein (AAK) Theory, Helson-Szego Theorem and Strong Szego Limit Theorem. Each of these theorem states the equivalence of certain conditions. Actually, to each theorem we add one more equivalent condition related to the CMV inverse scattering problem.

Motivation & Objective

• To resolve the inverse scattering problem for CMV matrices in the Szegö class, where the scattering function generally fails to uniquely determine the matrix due to missing singular measure information.
• To identify necessary and sufficient conditions under which the scattering function uniquely determines the CMV matrix, particularly in the absence of singular spectral components.
• To extend classical theorems (AAK, Helson-Szegö, Strong Szegö) by adding a new equivalent condition tied to the CMV inverse scattering problem.
• To establish a constructive framework for recovering CMV matrices from their scattering functions using GLM-type transformation operators and spectral theory on the unit circle.

Proposed method

• Utilizes the Faddeev-Marchenko inverse scattering framework adapted to CMV matrices, focusing on the generalized Gelfand-Levitan-Marchenko (GLM) operators that relate the standard basis to the intrinsic orthogonal basis of the perturbed system.
• Applies the Adamyan-Arov-Kreín (AAK) theory to characterize the class of CMV matrices for which the scattering correspondence is injective, linking approximation theory to spectral reconstruction.
• Incorporates the Helson-Szegö Theorem to define a subclass ${\mathbf{HS}}$ of CMV matrices where the scattering function determines the Verblunsky coefficients uniquely, under a condition on the spectral measure's logarithmic integral.
• Employs the Strong Szegö Limit Theorem to characterize a regular subclass ${\mathbf{Sz}}^{\text{reg}}$ where the scattering function uniquely determines the matrix, with a condition involving the convergence of $\sum n|a_n|^2$.
• Derives a factorization formula for the limit of the resolvent $\lim_{r\uparrow 1}(I - r^2{\mathcal{H}}^*{\mathcal{H}})^{-1}$, which leads to a determinant identity involving the products of spectral parameters $\rho_n$.
• Establishes a connection between the reproducing kernel in the subspace $\check{M}_s^+$ and the orthonormal basis $\{{\mathfrak{f}}_n\}$, enabling the construction of the transformation matrix $\mathcal{L}$ via triangular factorization.

Experimental results

Research questions

• RQ1 Under what conditions on the Verblunsky coefficients is the scattering function of a CMV matrix in the Szegö class sufficient to uniquely reconstruct the matrix?
• RQ2 How can the classical AAK, Helson-Szegö, and Strong Szegö theorems be extended to include a new equivalent condition related to the CMV inverse scattering problem?
• RQ3 What spectral and operator-theoretic conditions ensure that the generalized Gelfand-Levitan-Marchenko operator induces a one-to-one correspondence between CMV matrices and their scattering functions?
• RQ4 Can Widom's formula for determinants of Hankel operators be proven using scattering-theoretic methods in the context of CMV matrices?

Key findings

• The paper identifies a subclass ${\mathbf{Sz}}^{\text{reg}}$ of the Szegö class where the scattering function uniquely determines the CMV matrix, providing a solution to the inverse scattering problem.
• The Helson-Szegö condition is shown to be equivalent to the requirement that the GLM operator ${\mathcal{M}}$ is bounded, defining the class ${\mathbf{HS}} \subset {\mathbf{Sz}}^{\text{reg}}$.
• The Strong Szegö Limit Theorem is extended to include the condition $\sum n|a_n|^2 < \infty$, which ensures the injectivity of the scattering map and characterizes a regular subclass of CMV matrices.
• A new proof of Widom's formula is established via the factorization of the resolvent $\lim_{r\uparrow 1}(I - r^2{\mathcal{H}}^*{\mathcal{H}})^{-1} = {\mathcal{L}}{\mathcal{L}}^*$, linking it to the product of spectral parameters $\rho_n$.
• The determinant identity $\det(I - {\mathcal{H}}^*{\mathcal{H}}) = \prod_{n=0}^\infty \rho_n^{2(n+1)}$ is proven under the Hilbert-Schmidt condition on the Hankel operator ${\mathcal{H}}$, generalizing earlier trace-class results.
• The orthonormal basis $\{{\mathfrak{f}}_n\}$ in $\check{M}_s^+$ is constructed via the reproducing kernel $\check{\mathcal{K}}_0^{st^n}$, and its matrix representation $\mathcal{L}$ satisfies $\mathcal{L}^n_n = 1/(\rho_n \rho_{n+1} \cdots)$, enabling explicit reconstruction.