Scattering the geometry of weighted graphs

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.

References in corpus (3)

• Spectral Properties of Schrödinger Operators on Perturbed Lattices
• Spectral and scattering theory for Schrödinger operators on perturbed topological crystals
• Spectral and scattering theory for Gauss-Bonnet operators on perturbed topological crystals

Cited by in corpus (1)

• Spectral and scattering theory for topological crystals perturbed by infinitely many new edges