Trace formulas for Schr\"odinger operators on the half-line

TL;DR

This paper derives trace formulas and identities for Schrödinger operators on the half-line with Robin boundary conditions, linking resolvent differences to Wronskians and perturbation determinants.

Contribution

It introduces new trace formulas and identities for Schrödinger operators with Robin boundary conditions on the half-line, expanding the theoretical framework.

Findings

Derived an explicit expression for the resolvent difference trace

Established a representation for the perturbation determinant

Formulated trace identities of Buslaev-Faddeev type

Abstract

We study the scattering problem for the Schr\"odinger equation on the half-line with Robin boundary condition at the origin. We derive an expression for the trace of the difference of the perturbed and unperturbed resolvent in terms of a Wronskian. This leads to a representation for the perturbation determinant and to trace identities of Buslaev-Faddeev type.