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Absolutely continuous spectrum of discrete Schrödinger operators with slowly oscillating potentials
Author(s) -
Kim Ahyoung,
Kiselev Alexander
Publication year - 2009
Publication title -
mathematische nachrichten
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.913
H-Index - 50
eISSN - 1522-2616
pISSN - 0025-584X
DOI - 10.1002/mana.200810754
Subject(s) - mathematics , eigenfunction , spectrum (functional analysis) , bounded function , continuous spectrum , absolute continuity , operator (biology) , corollary , interval (graph theory) , mathematical physics , infinity , mathematical analysis , combinatorics , eigenvalues and eigenvectors , quantum mechanics , physics , chemistry , biochemistry , repressor , transcription factor , gene
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)