Transfer matrices, hyperbolic geometry and absolutely continuous spectrum for some discrete Schrödinger operators on graphs (Q813940)

For a Schrödinger operator \(\Delta +q\) on graphs the authors derive the Green's function as the unique \(l^2\) solution to \[ (\Delta +q-\lambda) G(v,w;\lambda )=\delta _w(v) \] under suitable restrictions on the potential \(q\) on a graph. Using this approach, they establish certain bounds of \(G\) to derive the absolute continuity of the spectrum of the operator \(\Delta +q\). We state a typical result. Theorem . Let \(q\) be a potential on \(\mathbb Z_+\) such that \(q_k\rightarrow 0\) as \(k\rightarrow \infty\) and \(\sum^{\infty}_{k=0}| q_{k+1}-q_k| <\infty\). Then for all \(0<\mu <2\) and \(\varepsilon >0\) \[ \sup_{\lambda \in K_{\mu,\varepsilon}} d(i,G(0,0;\lambda))<\infty, \] where \(K_{\mu,\varepsilon}:= \{\lambda \in H:| \Re\;\lambda | \leq \mu\), \(0<\Im\;\lambda \leq \varepsilon \}\).