Properties of the Scattering Matrix and Dispersion Estimates for Jacobi Operators

TL;DR

This paper investigates the properties of the scattering matrix for Jacobi operators with summable moments, demonstrating that derivatives of the scattering matrix belong to the Wiener algebra, and uses this to enhance dispersive estimates in the resonant case.

Contribution

It establishes the membership of scattering matrix derivatives in the Wiener algebra for Jacobi operators with summable moments, leading to improved dispersive estimates.

Findings

Derivatives of the scattering matrix are in the Wiener algebra under certain conditions.

Improved dispersive decay estimates for the Jacobi equation in the resonant case.

Enhanced understanding of scattering properties for Jacobi operators.

Abstract

We show that for a Jacobi operator with coefficients whose (j+1)'th moments are summable the j'th derivative of the scattering matrix is in the Wiener algebra of functions with summable Fourier coefficients. We use this result to improve the known dispersive estimates with integrable time decay for the time dependent Jacobi equation in the resonant case.