On the spectrum of Schrödinger operators with a finitely supported potential on the \(d\)-regular tree (Q732089)

The authors characterize the spectrum of the discrete Schrödinger operator \[ Hf(x)=V(x)f(x)+\frac{1}{d}\sum_{e\in A_x} [f(t(e))-f(o(e))] \] in \(l^2(T_d)\), where \(T_d\) is a \(d\)-regular tree, \(A_x\) is the set of all oriented edges of \(T_d\) originating at the vertex \(x\), and \(t(e)\) (resp. \(o(e)\)) denote the terminus (resp. origin) of the oriented edge \(e\). Fixing \(v\in \mathbb{R}\) and \(l\) a non-negative integer, for each vertex \(x\) the potential \(V(x)\) is assumed to \(1+v\) for \(d(o,x)=l\) and \(1\) otherwise, where \(o\) is the origin of \(T_d\) and \(d(o,x)\) is the graph distance from \(o\) to \(x\). Denoting the vertex set of \(T_d\) by \({\mathcal V}\), the Hilbert space \(l^2(T_d)\) is given by \[ l^2(T_d) =\left\{f:{\mathcal V}\to \mathbb{C}\;\left|\;\sum_{x\in{\mathcal V}} \frac{1}{d} |f(x)|^2<\infty\right\}\right.. \]