Trace Formulas for Schroedinger Operators in Connection with Scattering Theory for Finite-Gap Backgrounds

TL;DR

This paper develops trace formulas for one-dimensional Schrödinger operators with finite-gap backgrounds, linking spectral data, conserved quantities, and scattering theory using advanced mathematical tools like Abelian integrals.

Contribution

It introduces new trace formulas connecting spectral shift functions, reflection coefficients, and conserved quantities for finite-gap Schrödinger operators.

Findings

Established trace formulas relating spectral data and scattering coefficients.

Connected conserved quantities of KdV solutions to spectral and scattering data.

Utilized Abelian integrals on hyperelliptic Riemann surfaces to analyze operators.

Abstract

We investigate trace formulas for one-dimensional Schroedinger operators which are trace class perturbations of quasi-periodic finite-gap operators using Krein's spectral shift theory. In particular, we establish the conserved quantities for the solutions of the Korteweg-de Vries hierarchy in this class and relate them to the reflection coefficients via Abelian integrals on the underlying hyperelliptic Riemann surface.