Scattering the Geometry of Weighted Graphs


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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  1. 1.

    st-limt→± stands for the strong limit.


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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.


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).


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 \(\mathscr{L}(\mathscr{H},\widetilde {\mathscr{H}})\).


  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).

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  • Graphs
  • Laplacian
  • Scattering theory

Mathematics Subject Classification (2010)

  • 35P25
  • 05C63
  • 35P05