Explicit formula for Schroedinger wave operators on the half-line for potentials up to optimal decay

TL;DR

This paper derives an explicit formula for Schrödinger wave operators on the half-line with rapidly decaying potentials, combining the scattering operator, dilation generator, and a Hilbert-Schmidt remainder, with applications to Levinson's theorem.

Contribution

It provides a new explicit formula for wave operators on the half-line with optimal decay potentials, using elementary Fourier transform constructions.

Findings

Explicit wave operator formula involving scattering operator and dilation generator

Hilbert-Schmidt remainder term established

Topological interpretation of Levinson's theorem via index theory

Abstract

We give an explicit formula for the wave operators for Schroedinger operators on the half-line with a potential decaying strictly faster than the polynomial of degree minus two. The formula consists of the main term given by the scattering operator and a function of the generator of the dilation group, and a Hilbert-Schmidt remainder term. Our method is based on the elementary construction of the generalized Fourier transform in terms of the solutions of the Volterra integral equations. As a corollary, a topological interpretation of Levinson's theorem is established via an index theorem approach.