# Schroedinger wave operators on the discrete half-line

## TL;DR

This paper derives an explicit formula for Schroedinger wave operators on the discrete half-line, interpreting them as pseudo-differential operators and providing a topological perspective on Levinson's theorem linking phase shifts and bound states.

## Contribution

It introduces a new explicit formula for wave operators on the discrete half-line and offers a topological interpretation of Levinson's theorem.

## Key findings

- Wave operators expressed as one-dimensional pseudo-differential operators
- Topological interpretation of Levinson's theorem
- Connection between scattering phase shift and bound states

## Abstract

An explicit formula for the wave operators associated with Schroedinger operators on the discrete half-line is deduced from their stationary expressions. The formula enables us to understand the wave operators as one dimensional pseudo-differential operators of order zero. As an application, we give a topological interpretation for Levinson's theorem, which relates the scattering phase shift and the number of bound states of the system.