Levinson's theorem for graphs

TL;DR

This paper establishes an analog of Levinson's theorem for quantum scattering on a weighted finite graph with an attached semi-infinite path, relating bound states to the phase winding of the reflection coefficient.

Contribution

It introduces a Levinson's theorem analog for graph scattering, linking bound states to the phase of the reflection coefficient in a novel graph setting.

Findings

Number of bound states equals m minus half the phase winding

Half-bound states are counted as half a bound state

The theorem extends Levinson's result to graph-based scattering

Abstract

We prove an analog of Levinson's theorem for scattering on a weighted (m+1)-vertex graph with a semi-infinite path attached to one of its vertices. In particular, we show that the number of bound states in such a scattering problem is equal to m minus half the winding number of the phase of the reflection coefficient (where each so-called half-bound state is counted as half a bound state).