On the spectral theory and dispersive estimates for a discrete Schrödinger equation in one dimension

Based on the recent work [Komech et al., “Dispersive estimates for 1D discrete Schrödinger and Klein-Gordon equations,” Appl. Anal. 85, 1487 (2006)] for compact potentials, we develop the spectral theory for the one-dimensional discrete Schrödinger operator, Hϕ=(−Δ+V)ϕ=−(ϕn+1+ϕn−1−2ϕn)+Vnϕn. We show that under appropriate decay conditions on the general potential (and a nonresonance condition at the spectral edges), the spectrum of H consists …