# 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.

## References

- 1.Ando, K., Isozaki, H., Morioka, H.: Spectral properties of schrödinger operators on perturbed lattices. Ann. Henri Poincaré
**17**(8), 2103–2171 (2016)ADSMathSciNetCrossRefGoogle Scholar - 2.Bei, F., Güneysu, B., Müller, J.: Scattering theory of the Hodge-Laplacian under a conformal perturbation. J. Spectr. Theory
**7**(1), 235–267 (2017)MathSciNetCrossRefGoogle Scholar - 3.Breuer, J., Last, Y.: Stability of spectral types for Jacobi matrices under decaying random perturbations. J. Funct. Anal.
**245**(1), 249–283 (2007)MathSciNetCrossRefGoogle Scholar - 4.Colin de Verdière, Y., Truc, F.: Scattering theory for graphs isomorphic to a regular tree at infinity. J. Math. Phys.
**54**(6), 063502, 24pp (2013)MathSciNetCrossRefGoogle Scholar - 5.Deift, P., Killip, R.: On the absolutely continuous spectrum of one-dimensional schrödinger operators with square summable potentials. Comm. Math. Phys.
**203**, 341–347 (1999)ADSMathSciNetCrossRefGoogle Scholar - 6.Demuth, M.: On Topics in Spectral and Stochastic Analysis for Schrödinger Operators. Recent Developments in Quantum Mechanics (Poiana Brasov, 1989), vol. 12, pp 223–242. Math. Phys Stud., Kluwer Acad. Publ., Dordrecht (1991)Google Scholar
- 7.Demuth, M., Stollmann, P., Stolz, G., van Casteren, J.: Trace norm estimates for products of integral operators and diffusion semigroups. Integr. Equ. Oper. Theory
**23**(2), 145–153 (1995)MathSciNetCrossRefGoogle Scholar - 8.Güneysu, B., Thalmaier, A.: Scattering theory without injectivity radius assumptions and spectral stability for the Ricci flow. arXiv:1709.01612
- 9.Hempel, R., Post, O., Weder, R.: On open scattering channels for manifolds with ends. J. Funct. Anal.
**266**(9), 5526–5583 (2014)MathSciNetCrossRefGoogle Scholar - 10.Hempel, R., Post, O.: On Open Scattering Channels for a Branched Covering of the Euclidean Plane. arXiv:1712.09147 (2017)
- 11.Higuchi, Y., Nomura, Y.: Spectral structure of the Laplacian on a covering graph. Eur. J. Combin.
**30**(2), 570–585 (2009)MathSciNetCrossRefGoogle Scholar - 12.Keller, M.: Absolutely continuous spectrum for multi-type Galton Watson trees. Ann. Henri Poincare
**13**, 1745–1766 (2012)ADSMathSciNetCrossRefGoogle Scholar - 13.Keller, M., Lenz, D., Warzel, S.: On the spectral theory of trees with finite cone type. Israel J. Math.
**194**, 107–135 (2013)MathSciNetCrossRefGoogle Scholar - 14.Keller, M., Lenz, D., Warzel, S.: An invitation to trees of finite cone type: random and deterministic operators. Markov Process Relat Fields
**21**(3), 557–574 (2015). part 1MathSciNetGoogle Scholar - 15.Killip, R.: Perturbations of one-dimensional schrödinger operators preserving the absolutely continuous spectrum. Int. Math. Res. Not.
**38**, 2029–2061 (2002)MathSciNetCrossRefGoogle Scholar - 16.Kiselev, A.: Absolutely continuous spectrum of one-dimensional schrödinger operators and Jacobi matrices with slowly decreasing potentials. Comm. Math. Phys.
**179**, 377–400 (1996)ADSMathSciNetCrossRefGoogle Scholar - 17.Klein, A.: Absolutely continuous spectrum in the Anderson model on the Bethe lattice. Math. Res. Lett.
**1**, 399–407 (1994)MathSciNetCrossRefGoogle Scholar - 18.Last, Y., Simon, B.: Eigenfunctions, transfer matrices, and absolutely continuous spectrum of onedimensional Schrödinger operators. Invent. Math.
**135**, 329–367 (1999)ADSMathSciNetCrossRefGoogle Scholar - 19.Müller, W., Salomonsen, G.: Scattering theory for the Laplacian on manifolds with bounded curvature. J. Funct. Anal.
**253**(1), 158–206 (2007)MathSciNetCrossRefGoogle Scholar - 20.Nagnibeda, T., Woess, W.: Random walks on trees with finite cone type. J. Theoret. Probab.
**15**, 383–422 (2002)MathSciNetCrossRefGoogle Scholar - 21.Parra, D.: Spectral and scattering theory for Gauss-Bonnet operators on perturbed topological crystals. J. Math. Anal. Appl.
**452**(2), 792–813 (2017)MathSciNetCrossRefGoogle Scholar - 22.Parra, D., Richard, S.: Spectral and scattering theory for Schroedinger operators on perturbed topological crystals. Rev. Math. Phys.
**30**, 1850009–1 - 1850009-39 (2018)CrossRefGoogle Scholar - 23.Reed, M., Simon, B.: Methods of Modern Mathematical Physics. III. Scattering Theory. Academic Press [Harcourt Brace Jovanovich Publishers], New York (1979)zbMATHGoogle Scholar
- 24.Remling, C.: The absolutely continuous spectrum of one-dimensional schrödinger operators with decaying potentials. Comm. Math. Phys.
**193**, 151–170 (1998)ADSMathSciNetCrossRefGoogle Scholar - 25.Stollmann, P.: Scattering by obstacles of finite capacity. J. Funct. Anal.
**121**(2), 416–425 (1994)MathSciNetCrossRefGoogle Scholar

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