Scattering the geometry of weighted graphs

TL;DR

This paper establishes a weighted $L^1$-criterion for the existence and completeness of wave operators between two weighted graphs, providing a broad condition for their absolutely continuous spectra to coincide.

Contribution

It introduces a new weighted $L^1$-criterion for wave operators on weighted graphs, linking graph geometry to spectral properties.

Findings

Wave operators exist and are complete under the criterion.

Absolutely continuous spectra of the graphs are equal under the conditions.

Provides a general framework connecting graph weights and spectral theory.

Abstract

Given two weighted graphs $(X,b_k,m_k)$, $k=1,2$ with $b_1\sim b_2$ and $m_1\sim m_2$, we prove a weighted $L^1$-criterion for the existence and completeness of the wave operators $ W_{\pm}(H_{2},H_1, I_{1,2})$, where $H_k$ denotes the natural Laplacian in $\ell^2(X,m_k)$ w.r.t. $(X,b_k,m_k)$ and $I_{1,2}$ the trivial identification of $\ell^2(X,m_1)$ with $\ell^2(X,m_2)$. In particular, this entails a very general criterion for the absolutely continuous spectra of $H_1$ and $H_2$ to be equal.