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.
This is a preview of subscription content, access via your institution.
Notes
 1.
stlim_{t→±∞} stands for the strong limit.
References
 1.
Ando, K., Isozaki, H., Morioka, H.: Spectral properties of schrödinger operators on perturbed lattices. Ann. Henri Poincaré 17(8), 2103–2171 (2016)
 2.
Bei, F., Güneysu, B., Müller, J.: Scattering theory of the HodgeLaplacian under a conformal perturbation. J. Spectr. Theory 7(1), 235–267 (2017)
 3.
Breuer, J., Last, Y.: Stability of spectral types for Jacobi matrices under decaying random perturbations. J. Funct. Anal. 245(1), 249–283 (2007)
 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)
 5.
Deift, P., Killip, R.: On the absolutely continuous spectrum of onedimensional schrödinger operators with square summable potentials. Comm. Math. Phys. 203, 341–347 (1999)
 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)
 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)
 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)
 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)
 12.
Keller, M.: Absolutely continuous spectrum for multitype Galton Watson trees. Ann. Henri Poincare 13, 1745–1766 (2012)
 13.
Keller, M., Lenz, D., Warzel, S.: On the spectral theory of trees with finite cone type. Israel J. Math. 194, 107–135 (2013)
 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 1
 15.
Killip, R.: Perturbations of onedimensional schrödinger operators preserving the absolutely continuous spectrum. Int. Math. Res. Not. 38, 2029–2061 (2002)
 16.
Kiselev, A.: Absolutely continuous spectrum of onedimensional schrödinger operators and Jacobi matrices with slowly decreasing potentials. Comm. Math. Phys. 179, 377–400 (1996)
 17.
Klein, A.: Absolutely continuous spectrum in the Anderson model on the Bethe lattice. Math. Res. Lett. 1, 399–407 (1994)
 18.
Last, Y., Simon, B.: Eigenfunctions, transfer matrices, and absolutely continuous spectrum of onedimensional Schrödinger operators. Invent. Math. 135, 329–367 (1999)
 19.
Müller, W., Salomonsen, G.: Scattering theory for the Laplacian on manifolds with bounded curvature. J. Funct. Anal. 253(1), 158–206 (2007)
 20.
Nagnibeda, T., Woess, W.: Random walks on trees with finite cone type. J. Theoret. Probab. 15, 383–422 (2002)
 21.
Parra, D.: Spectral and scattering theory for GaussBonnet operators on perturbed topological crystals. J. Math. Anal. Appl. 452(2), 792–813 (2017)
 22.
Parra, D., Richard, S.: Spectral and scattering theory for Schroedinger operators on perturbed topological crystals. Rev. Math. Phys. 30, 1850009–1  185000939 (2018)
 23.
Reed, M., Simon, B.: Methods of Modern Mathematical Physics. III. Scattering Theory. Academic Press [Harcourt Brace Jovanovich Publishers], New York (1979)
 24.
Remling, C.: The absolutely continuous spectrum of onedimensional schrödinger operators with decaying potentials. Comm. Math. Phys. 193, 151–170 (1998)
 25.
Stollmann, P.: Scattering by obstacles of finite capacity. J. Funct. Anal. 121 (2), 416–425 (1994)
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.
Author information
Affiliations
Corresponding author
Appendices
Appendix A: Belopol’skiiBirman Theorem
Theorem B.1
(Belopol’skiiBirman) For k = 1, 2, let \(H_{k}\geqslant 0\) be a selfadjoint operatorin a complex Hilbert space \(\mathscr{H}_{k}\), where in the sequel π_{ac}(H_{k}) denotes the projection onto the H_{k}absolutely continuous subspace of \(\mathscr{H}_{k}\). Assume that \(I\in \mathscr{L}(\mathscr{H}_{1}, \mathscr{H}_{2})\) is such that the following assumptions hold:

I has a twosided bounded inverse

One has either \(I(\text {Dom}(\sqrt {H_{1}}))=\text {Dom}(\sqrt {H_{2}})\) orI(Dom(H_{1})) = Dom(H_{2}).

The operator
$$\begin{array}{@{}rcl@{}} (I^{*}I\text{id}_{\mathscr{H}_{1}})\exp(r H_{1}):\mathscr{H}_{1}\to\mathscr{H}_{1} \>\>\text{ is compact for some }r>0. \end{array} $$ 
There exists a trace class operator \(T:\mathscr{H}_{1}\to \mathscr{H}_{2}\) anda number s > 0 suchthat for all f_{2} ∈Dom(H_{2}),f_{1} ∈Dom(H_{1}) one has
$$\begin{array}{@{}rcl@{}} \left\langle f_{2} ,Tf_{1}\right\rangle_{\mathscr{H}_{2}}\>=\>&\left\langle H_{2}f_{2}, \exp(sH_{2}) I \exp(sH_{1})f_{1}\right\rangle_{\mathscr{H}_{2}} \\ &\left\langle f_{2}, \exp(sH_{2}) I \exp(sH_{1}) H_{1}f_{1}\right\rangle_{\mathscr{H}_{2}}. \end{array} $$
Then the wave operators
exist^{Footnote 1} and are complete, where completeness means that
Moreover, W_{±}(H_{2}, H_{1}, I) are partial isometries with inital space Ranπ_{ac}(H_{1}) and final space Ranπ_{ac}(H_{2}), and one has spec_{ac}(H_{1}) = spec_{ac}(H_{2}).
Proof
This result is essentially included in Theorem XI.13 from [23]. A detailed proof is given in [8]. □
Appendix B: Some Facts on Quadratic forms in Hilbert Spaces
The following result is certainly wellknown. As we have not been able to find a precise reference, we have included a quick proof:
Proposition B.1
Let D be a densely defined, closed operator from a Hilbert space \(\mathscr{H}\) to another Hilbert space \(\widetilde {\mathscr{H}}\). Then the following assertions hold:

a)
The nonnegative, densely defined sesquilinear formQ_{D} in \(\mathscr{H}\) givenby
$$Q_{D}(f):= \left\Df\right\^{2}, \quad \text{Dom} (Q_{D})= \text{Dom} (D), $$is closed, and its associated nonnegative selfadjoint operator is D^{∗}D.

b)
For all t > 0 the operator D exp(−tD^{∗}D) from \(\mathscr{H}\) to \(\widetilde {\mathscr{H}}\) is in \(\mathscr{L}(\mathscr{H},\widetilde {\mathscr{H}})\).
Proof

a)
It is checked easily that Q_{D} is closed. Let \(H_{D}\geqslant 0\) denote its associated selfadjoint operator. If f_{1} ∈Dom(H_{D}), then we have f_{1} ∈Dom(Q_{D}) = Dom(D), and for all f_{2} ∈Dom(D),
$$\left\langle H_{D}f_{1}, f_{2}\right\rangle=Q_{D}(f_{1},f_{2})=\left\langle Df_{1}, Df_{2}\right\rangle, $$which implies Df_{1} ∈Dom(D^{∗}) and D^{∗}Df_{1} = H_{D}f_{1}. Conversely, if f_{1} ∈Dom(D) = Dom(Q_{D}) with Df_{1} ∈Dom(D^{∗}), then for all f_{2} ∈Dom(D) = Dom(Q_{D}) we have
$$Q_{D}(f_{1},f_{2})=\left\langle Df_{1}, Df_{2}\right\rangle = \left\langle D^{*} Df_{1}, f_{2}\right\rangle, $$which implies f_{1} ∈Dom(H_{D}) and H_{D}f_{1} = D^{∗}Df_{1}.

b)
Set H_{D} := D^{∗}D. The polar decomposition of D reads \(D= U\sqrt {H_{D}}\), where U is an everywhere defined operator from \(\mathscr{H}\) to \(\widetilde {\mathscr{H}}\) which maps
$$\overline{\text{Ran}(\sqrt{H_{D}})}\longrightarrow \overline{\text{Ran}(D)}\quad\text{ isometrically}. $$Thus, we have
$$\left\D\exp(t D^{*}D)\right\=\left\U\sqrt{H_{D}} \exp(t H_{D})\right\=\left\\sqrt{H_{D}} \exp(t H_{D})\right\, $$which is < ∞ for all t > 0 by the spectral calculus.
□
Rights and permissions
About this article
Cite this article
Güneysu, B., Keller, M. Scattering the Geometry of Weighted Graphs. Math Phys Anal Geom 21, 28 (2018). https://doi.org/10.1007/s1104001892851
Received:
Accepted:
Published:
Keywords
 Graphs
 Laplacian
 Scattering theory
Mathematics Subject Classification (2010)
 35P25
 05C63
 35P05