Advertisement

Scattering the Geometry of Weighted Graphs

  • Batu Güneysu
  • Matthias Keller
Article
  • 19 Downloads

Abstract

Given two weighted graphs (X, bk, mk), k = 1,2 with b1b2 and m1m2, we prove a weighted L1-criterion for the existence and completeness of the wave operators W±(H2, H1, I1,2), where Hk denotes the natural Laplacian in 2(X, mk) w.r.t. (X, bk, mk) and I1,2 the trivial identification of 2(X, m1) with 2(X, m2). In particular, this entails a general criterion for the absolutely continuous spectra of H1 and H2 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. 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. 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. 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. 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. 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. 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. 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. 8.
    Güneysu, B., Thalmaier, A.: Scattering theory without injectivity radius assumptions and spectral stability for the Ricci flow. arXiv:1709.01612
  9. 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. 10.
    Hempel, R., Post, O.: On Open Scattering Channels for a Branched Covering of the Euclidean Plane. arXiv:1712.09147 (2017)
  11. 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. 12.
    Keller, M.: Absolutely continuous spectrum for multi-type Galton Watson trees. Ann. Henri Poincare 13, 1745–1766 (2012)ADSMathSciNetCrossRefGoogle Scholar
  13. 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. 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. 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. 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. 17.
    Klein, A.: Absolutely continuous spectrum in the Anderson model on the Bethe lattice. Math. Res. Lett. 1, 399–407 (1994)MathSciNetCrossRefGoogle Scholar
  18. 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. 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. 20.
    Nagnibeda, T., Woess, W.: Random walks on trees with finite cone type. J. Theoret. Probab. 15, 383–422 (2002)MathSciNetCrossRefGoogle Scholar
  21. 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. 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. 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. 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. 25.
    Stollmann, P.: Scattering by obstacles of finite capacity. J. Funct. Anal. 121 (2), 416–425 (1994)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Institut für MathematikHumboldt-Universität zu BerlinBerlinGermany
  2. 2.Institut für MathematikUniversität PotsdamPotsdamGermany

Personalised recommendations