D.E Pelinovsky and A. Stefanov

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

Abstract:
Based on the recent work for compact potentials, we develop the spectral theory for the one-dimensional discrete Schrodinger operator

H φ = (-Δ + V)φ = -(φ_n+1 + φ_n-1 - 2 φ_n) + V_n φ_n.

We show that under appropriate decay conditions on the general potential (and a non-resonance 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

| e^{i t H} P_a.c.(H) |_{l²_σ ->
l²_-σ} < t^-3/2

for any fixed σ > 5/2 and any t > 0, 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, we find new dispersive estimates

| e^{i t H} P_a.c.(H) |_{l¹ ->
l^inf} < t^-1/3

These estimates are sharp for the discrete Schrodinger operators even for V = 0.

Keywords:
Discrete nonlinear Schrodinger equation, limiting absorption principle, Puiseux expansions, discrete Jost functions, dispersive estimates.