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Levinson's theorem for graphs

### 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).

© 2011 American Institute of Physics

Received 05 April 2011
Accepted 10 July 2011
Published online 11 August 2011

Acknowledgments:
We thank Jeffrey Goldstone for sharing his proof of completeness of the scattering and bound states^{8} and Gorjan Alagic, Aaron Denney, and Cris Moore for discussions of methods for computing *S*-matrices of graphs. We also thank an anonymous referee for pointing out several corrections to an earlier version of this paper. This work was supported in part by MITACS, NSERC, QuantumWorks, and the US ARO/DTO.

Article outline:

I. INTRODUCTION
II. SCATTERING ON GRAPHS
A. Scattering states
B. Bound states
III. LEVINSON'S THEOREM
IV. DISCUSSION

/content/aip/journal/jmp/52/8/10.1063/1.3622608

http://aip.metastore.ingenta.com/content/aip/journal/jmp/52/8/10.1063/1.3622608

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