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Absolutely continuous spectrum of discrete Schrödinger operators with slowly oscillating potentials

✍ Scribed by Ahyoung Kim; Alexander Kiselev

Publisher
John Wiley and Sons
Year
2009
Tongue
English
Weight
220 KB
Volume
282
Category
Article
ISSN
0025-584X

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✦ Synopsis

Abstract

We show that when a potential b~n~ of a discrete Schrödinger operator, defined on l^2^(ℤ^+^), slowly oscillates satisfying the conditions b~n~ ∈ l^∞^ and ∂b~n~ = b~n +1~ – b~n~ ∈ l^p^, p < 2, then all solutions of the equation Ju = Eu are bounded near infinity at almost every E ∈ [–2 + lim sup~n →∞~ b~n~, 2 + lim sup~n →∞~ b~n~ ] ∩ [–2 + lim inf~n →∞~ b~n~, 2 + lim inf~n →∞~ b~n~ ]. We derive an asymptotic formula for generalized eigenfunctions in this case. As a corollary, the absolutely continuous spectrum of the corresponding Jacobi operator is essentially supported on the same interval of E (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

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