Inverse scattering problem for a special class of canonical systems and non-linear Fourier integral. Part I: asymptotics of eigenfunctions

TL;DR

This paper develops an approach to the inverse scattering problem for a class of canonical systems, extending previous work to include pure point spectra and generalizing Sturm-Liouville operators, with focus on eigenfunction asymptotics.

Contribution

It introduces a novel method for inverse scattering in canonical systems with mixed spectra, broadening the scope beyond Sturm-Liouville operators.

Findings

Solution of inverse scattering for a class of canonical systems.

Asymptotic analysis of eigenfunctions.

Extension to systems with pure point spectrum.

Abstract

An original approach to the inverse scattering for Jacobi matrices was suggested in a recent paper by Volberg-Yuditskii. The authors considered quite sophisticated spectral sets (including Cantor sets of positive Lebesgue measure), however they did not take into account the mass point spectrum. This paper follows similar lines for the continuous setting with an absolutely continuous spectrum on the half-axis and a pure point spectrum on the negative half-axis satisfying the Blaschke condition. This leads us to the solution of the inverse scattering problem for a class of canonical systems that generalizes the case of Sturm-Liouville (Schr\"odinger) operator.