Ground state energy of trimmed discrete Schrödinger operators and localization for trimmed Anderson models (Q402289)

The authors consider discrete Schrödinger operators of the form \(H=-\Delta+V\) on \(l^2(\mathbb Z^d),\) where \(\Delta\) is the discrete Laplacian and \(V\) is a bounded potential. Given \(\Gamma\nsubseteq \mathbb Z^d,\) the \(\Gamma\)-trimming is the restriction \(H_\Gamma\) of \(\chi_{\Gamma^c}H\chi_{\Gamma^c}\) to \(l^2(\Gamma^c),\) where \(\chi_A\) denotes the characteristic function of the set \(A\) and \(A^c= \mathbb Z^d\backslash A\) for \(A\subset \mathbb Z^d.\) It is proved that for relatively dense proper subsets \(\Gamma\) of \(\mathbb Z^d\) there exists strict lifting of the bottom of the spectrum, i.e., \(E_\Gamma(H)>E_\emptyset(H)\). Wegner estimates are established as well as a localization at the bottom of the spectrum for \(\Gamma\)-trimmed Anderson models, i.e., Anderson models with random potential supported by the set \(\Gamma\).