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

Journal Article · Sat Nov 15 04:00:00 EST 2008 · Journal of Mathematical Physics

DOI:https://doi.org/10.1063/1.3005597· OSTI ID:21175776

Pelinovsky, D E ^[1]; Stefanov, A ^[2]

---

1. Department of Mathematics, McMaster University, Hamilton, Ontario L8S 4K1 (Canada)
2. Department of Mathematics, University of Kansas, 1460 Jayhawk Blvd., Lawrence, Kansas 66045-7523 (United States)

Based on the recent work [Komech et al., 'Dispersive estimates for 1D discrete Schroedinger and Klein-Gordon equations', Appl. Anal. 85, 1487 (2006)] for compact potentials, we develop the spectral theory for the one-dimensional discrete Schroedinger operator, H{phi}=(-{delta}+V){phi}=-({phi}{sub n+1}+{phi}{sub n-1}-2{phi}{sub n})+V{sub n}{phi}{sub 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 e{sup itH}P{sub a.c.}(H) parallel {sub l{sub {sigma}{sup 2}}{yields}{sub l-{sigma}{sup 2}}} < or approx. t{sup -3/2} for any fixed {sigma}>(5/2) and any t>0, where P{sub 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 Schroedinger and Klein-Gordon equations,'' Nonlinearity 18, 1841 (2005)], we find new dispersive estimates parallel e{sup itH}P{sub a.c.}(H) parallel {sub l{sup 1}{yields}{sub l{sup {infinity}}}} < or approx. t{sup -1/3}, which are sharp for the discrete Schroedinger operators even for V=0.

• OSTI ID: 21175776
• Journal Information: Journal of Mathematical Physics, Journal Name: Journal of Mathematical Physics Journal Issue: 11 Vol. 49; ISSN JMAPAQ; ISSN 0022-2488
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
ASYMPTOTIC SOLUTIONS
EIGENFUNCTIONS
EIGENVALUES
KLEIN-GORDON EQUATION
MULTIPLICITY
NONLINEAR PROBLEMS
ONE-DIMENSIONAL CALCULATIONS
POTENTIALS
SCHROEDINGER EQUATION