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

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    StefanovA_JMP_49(11)113501.pdf (177.3Kb)
    Issue Date
    2008-11-08
    Author
    Pelinovsky, Dmitry E.
    Stefanov, Atanas G.
    Publisher
    American Institute of Physics
    Type
    Article
    Article Version
    Scholarly/refereed, publisher version
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    Abstract
    Based on the recent work [Komech et al., “Dispersive estimates for 1D discrete Schrödinger and Klein-Gordon equations,” Appl. Anal.85, 1487 (2006)] for compact potentials, we develop the spectral theory for the one-dimensional discrete Schrödinger operator, Hϕ=(−Δ+V)ϕ=−(ϕn+1+ϕn−1−2ϕn)+Vnϕn. 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 ∥eitHPa.c.(H)∥l2σ→l2−σ≲t−3/2 for any fixed σ>52 and any t>0, where Pa.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 Schrödinger and Klein-Gordon equations,” Nonlinearity18, 1841 (2005)], we find new dispersive estimates ∥eitHPa.c.(H)∥l1→l∞≲t−1/3, which are sharp for the discrete Schrödinger operators even for V=0.
    Description
    This is the published version, also available here: http://dx.doi.org/10.1063/1.3005597.
    URI
    http://hdl.handle.net/1808/17221
    DOI
    https://doi.org/10.1063/1.3005597
    Collections
    • Mathematics Scholarly Works [282]
    Citation
    Pelinovsky, D. E. & Stefanov, A. "On the spectral theory and dispersive estimates for a discrete Schrödinger equation in one dimension." J. Math. Phys. 49, 113501 (2008); http://dx.doi.org/10.1063/1.3005597.

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    KU Libraries
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    Lawrence, KS 66045
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    Contact KU ScholarWorks
    785-864-8983
    KU Libraries
    1425 Jayhawk Blvd
    Lawrence, KS 66045
    785-864-8983

    KU Libraries
    1425 Jayhawk Blvd
    Lawrence, KS 66045
    Image Credits
     

     

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