# Scattering the Geometry of Weighted Graphs

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## Abstract

Given two weighted graphs (*X*, *b*_{k}, *m*_{k}), *k* = 1,2 with *b*_{1} ∼ *b*_{2} and *m*_{1} ∼ *m*_{2}, we prove a weighted *L*^{1}-criterion for the existence and completeness of the wave operators *W*_{±}(*H*_{2}, *H*_{1}, *I*_{1,2}), where *H*_{k} denotes the natural Laplacian in *ℓ*^{2}(*X*, *m*_{k}) w.r.t. (*X*, *b*_{k}, *m*_{k}) and *I*_{1,2} the trivial identification of *ℓ*^{2}(*X*, *m*_{1}) with *ℓ*^{2}(*X*, *m*_{2}). In particular, this entails a general criterion for the absolutely continuous spectra of *H*_{1} and *H*_{2} to be equal.

## Keywords

Graphs Laplacian Scattering theory## Mathematics Subject Classification (2010)

35P25 05C63 35P05## Notes

### Acknowledgments

The authors are grateful for various discussions and hints on the literature by Jonathan Breuer, Evgeny Korotyaev, Peter Stollmann and Francoise Truc. Furthermore, the second author acknowledges the support of this research by the DFG.

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