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

Based on the recent work [Komech , "Dispersive estimates for 1D discrete Schrodinger and Klein-Gordon equations," Appl. Anal. 85, 1487 (2006)] for compact potentials, we develop the spectral theory for the one-dimensional discrete Schrodinger operator, H phi=(-Delta+V)phi=-(phi(n+1)+phi(n-1)-2 phi(n))+V-n phi(n). We show that under appropriate decay conditions on the general potential (and a nonresonance condition at the spectral edges), the spectrum of H consists of finitely many eigenvalues of finite multiplicities and the essential (absolutely continuous) spectrum, while the resolvent satisfies the limiting absorption principle and the Puiseux expansions near the edges. These properties imply the dispersive estimates parallel to e(itH)P(a.c.)(H)parallel to(2)(l sigma)-> l(-sigma)(2)less than or similar to t(-3/2) for any fixed sigma>52t0, where P-a.c.(H) denotes the spectral projection to the absolutely continuous spectrum of H. In addition, based on the scattering theory for the discrete Jost solutions and the previous results by Stefanov and Kevrekidis ["Asymptotic behaviour of small solutions for the discrete nonlinear Schrodinger and Klein-Gordon equations," Nonlinearity 18, 1841 (2005)], we find new dispersive estimates parallel to e(itH)P(a.c.)(H)parallel to(1)(l)-> l(infinity)less than or similar to t(-1/3), which are sharp for the discrete Schrodinger operators even for V=0.