Scattering the Geometry of Weighted Graphs

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.

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Notes

  1. 1.

    st-limt→± stands for the strong limit.

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)

    ADS  MathSciNet  Article  Google 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)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  Google 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)

    ADS  MathSciNet  Article  Google 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)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Keller, M.: Absolutely continuous spectrum for multi-type Galton Watson trees. Ann. Henri Poincare 13, 1745–1766 (2012)

    ADS  MathSciNet  Article  Google 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)

    MathSciNet  Article  Google 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 1

    MathSciNet  Google Scholar 

  15. 15.

    Killip, R.: Perturbations of one-dimensional schrödinger operators preserving the absolutely continuous spectrum. Int. Math. Res. Not. 38, 2029–2061 (2002)

    MathSciNet  Article  Google 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)

    ADS  MathSciNet  Article  Google Scholar 

  17. 17.

    Klein, A.: Absolutely continuous spectrum in the Anderson model on the Bethe lattice. Math. Res. Lett. 1, 399–407 (1994)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Last, Y., Simon, B.: Eigenfunctions, transfer matrices, and absolutely continuous spectrum of onedimensional Schrödinger operators. Invent. Math. 135, 329–367 (1999)

    ADS  MathSciNet  Article  Google 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)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Nagnibeda, T., Woess, W.: Random walks on trees with finite cone type. J. Theoret. Probab. 15, 383–422 (2002)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  Google 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)

    Article  Google Scholar 

  23. 23.

    Reed, M., Simon, B.: Methods of Modern Mathematical Physics. III. Scattering Theory. Academic Press [Harcourt Brace Jovanovich Publishers], New York (1979)

    Google Scholar 

  24. 24.

    Remling, C.: The absolutely continuous spectrum of one-dimensional schrödinger operators with decaying potentials. Comm. Math. Phys. 193, 151–170 (1998)

    ADS  MathSciNet  Article  Google Scholar 

  25. 25.

    Stollmann, P.: Scattering by obstacles of finite capacity. J. Funct. Anal. 121 (2), 416–425 (1994)

    MathSciNet  Article  Google Scholar 

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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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Correspondence to Batu Güneysu.

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

$$W_{\pm}(H_{2},H_{1}, I)=\underset{t\to\pm\infty}{\text{st-lim}}\exp(itH_{2})I\exp(-itH_{1})\pi_{\text{ac}}(H_{1})$$

existFootnote 1 and are complete, where completeness means that

$$\left( \text{Ker} W_{\pm}(H_{2},H_{1}, I)\right)^{\perp}=\text{Ran} \pi_{\text{ac}}(H_{1}), \quad\overline{\text{Ran} W_{\pm}(H_{2},H_{1}, I)}=\text{Ran} \pi_{\text{ac}}(H_{2}).$$

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:

  1. 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 DD.

  2. b)

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

Proof

  1. 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 DDf1 = 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 = DDf1.

  2. b)

    Set HD := DD. 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.

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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

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Keywords

  • Graphs
  • Laplacian
  • Scattering theory

Mathematics Subject Classification (2010)

  • 35P25
  • 05C63
  • 35P05