Abstract
Given two weighted graphs (X, bk, mk), k = 1,2 with b1 ∼ b2 and m1 ∼ m2, 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 ℓ2(X, mk) w.r.t. (X, bk, mk) and I1,2 the trivial identification of ℓ2(X, m1) with ℓ2(X, m2). In particular, this entails a general criterion for the absolutely continuous spectra of H1 and H2 to be equal.
Similar content being viewed by others
Notes
st-limt→±∞ stands for the strong limit.
References
Ando, K., Isozaki, H., Morioka, H.: Spectral properties of schrödinger operators on perturbed lattices. Ann. Henri Poincaré 17(8), 2103–2171 (2016)
Bei, F., Güneysu, B., Müller, J.: Scattering theory of the Hodge-Laplacian under a conformal perturbation. J. Spectr. Theory 7(1), 235–267 (2017)
Breuer, J., Last, Y.: Stability of spectral types for Jacobi matrices under decaying random perturbations. J. Funct. Anal. 245(1), 249–283 (2007)
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)
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)
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)
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)
Güneysu, B., Thalmaier, A.: Scattering theory without injectivity radius assumptions and spectral stability for the Ricci flow. arXiv:1709.01612
Hempel, R., Post, O., Weder, R.: On open scattering channels for manifolds with ends. J. Funct. Anal. 266(9), 5526–5583 (2014)
Hempel, R., Post, O.: On Open Scattering Channels for a Branched Covering of the Euclidean Plane. arXiv:1712.09147 (2017)
Higuchi, Y., Nomura, Y.: Spectral structure of the Laplacian on a covering graph. Eur. J. Combin. 30(2), 570–585 (2009)
Keller, M.: Absolutely continuous spectrum for multi-type Galton Watson trees. Ann. Henri Poincare 13, 1745–1766 (2012)
Keller, M., Lenz, D., Warzel, S.: On the spectral theory of trees with finite cone type. Israel J. Math. 194, 107–135 (2013)
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
Killip, R.: Perturbations of one-dimensional schrödinger operators preserving the absolutely continuous spectrum. Int. Math. Res. Not. 38, 2029–2061 (2002)
Kiselev, A.: Absolutely continuous spectrum of one-dimensional schrödinger operators and Jacobi matrices with slowly decreasing potentials. Comm. Math. Phys. 179, 377–400 (1996)
Klein, A.: Absolutely continuous spectrum in the Anderson model on the Bethe lattice. Math. Res. Lett. 1, 399–407 (1994)
Last, Y., Simon, B.: Eigenfunctions, transfer matrices, and absolutely continuous spectrum of onedimensional Schrödinger operators. Invent. Math. 135, 329–367 (1999)
Müller, W., Salomonsen, G.: Scattering theory for the Laplacian on manifolds with bounded curvature. J. Funct. Anal. 253(1), 158–206 (2007)
Nagnibeda, T., Woess, W.: Random walks on trees with finite cone type. J. Theoret. Probab. 15, 383–422 (2002)
Parra, D.: Spectral and scattering theory for Gauss-Bonnet operators on perturbed topological crystals. J. Math. Anal. Appl. 452(2), 792–813 (2017)
Parra, D., Richard, S.: Spectral and scattering theory for Schroedinger operators on perturbed topological crystals. Rev. Math. Phys. 30, 1850009–1 - 1850009-39 (2018)
Reed, M., Simon, B.: Methods of Modern Mathematical Physics. III. Scattering Theory. Academic Press [Harcourt Brace Jovanovich Publishers], New York (1979)
Remling, C.: The absolutely continuous spectrum of one-dimensional schrödinger operators with decaying potentials. Comm. Math. Phys. 193, 151–170 (1998)
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
Authors and Affiliations
Corresponding author
Appendices
Appendix A: Belopol’skii-Birman Theorem
Theorem B.1
(Belopol’skii-Birman) For k = 1, 2, let \(H_{k}\geqslant 0\) be a self-adjoint operatorin a complex Hilbert space \(\mathscr{H}_{k}\), where in the sequel πac(Hk) denotes the projection onto the Hk-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 two-sided bounded inverse
-
One has either \(I(\text {Dom}(\sqrt {H_{1}}))=\text {Dom}(\sqrt {H_{2}})\) orI(Dom(H1)) = Dom(H2).
-
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 f2 ∈Dom(H2),f1 ∈Dom(H1) 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
existFootnote 1 and are complete, where completeness means that
Moreover, W±(H2, H1, I) are partial isometries with inital space Ranπac(H1) and final space Ranπac(H2), and one has specac(H1) = specac(H2).
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 well-known. 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 formQD 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 self-adjoint 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 QD is closed. Let \(H_{D}\geqslant 0\) denote its associated self-adjoint operator. If f1 ∈Dom(HD), then we have f1 ∈Dom(QD) = Dom(D), and for all f2 ∈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 Df1 ∈Dom(D∗) and D∗Df1 = HDf1. Conversely, if f1 ∈Dom(D) = Dom(QD) with Df1 ∈Dom(D∗), then for all f2 ∈Dom(D) = Dom(QD) 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 f1 ∈Dom(HD) and HDf1 = D∗Df1.
-
b)
Set HD := 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/s11040-018-9285-1
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11040-018-9285-1