Levinson's theorem for graphs II

TL;DR

This paper extends Levinson's theorem to complex graphs with multiple semi-infinite paths, relating bound states to the S-matrix's winding number and establishing completeness of bound and scattering states.

Contribution

It generalizes Levinson's theorem and the completeness proof to graphs with multiple semi-infinite paths, broadening the theoretical framework for quantum scattering on graphs.

Findings

Levinson's theorem is proven for (m+n)-vertex graphs with n semi-infinite paths.

The number of bound states is related to the winding of the S-matrix determinant.

Bound and scattering states form a complete basis for the Hilbert space.

Abstract

We prove Levinson's theorem for scattering on an (m+n)-vertex graph with n semi-infinite paths each attached to a different vertex, generalizing a previous result for the case n=1. This theorem counts the number of bound states in terms of the winding of the determinant of the S-matrix. We also provide a proof that the bound states and incoming scattering states of the Hamiltonian together form a complete basis for the Hilbert space, generalizing another result for the case n=1.