On the dispersion decay for crystals in the linearized Schrodinger-Poisson model

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:464

Article

Komech, A.^1,2,3  Kopylova, E.^1,2 

[1] Univ Vienna, Fac Math, Vienna, Austria

[2] RAS, Inst Informat Transmiss Problems, Moscow, Russia

[3] Moscow MV Lomonosov State Univ, Mech Math Dept, Moscow, Russia

关键词: Crystal;    Schrodinger-Poisson equations;    Ground state;    Bloch transform;    Positivity;    Dispersion decay;   

DOI  :  10.1016/j.jmaa.2018.04.035

来源: Elsevier

【 摘 要 】

The Schrodinger-Poisson-Newton equations for crystals with a cubic lattice and one ion per cell are considered. The ion charge density is assumed i) to satisfy the Wiener and Jellium conditions introduced in our previous paper [25], and ii) to be exponentially decaying at infinity. The corresponding examples are given. We study the linearized dynamics at the ground state. The dispersion relations are introduced via spectral resolution for the non-selfadjoint Hamilton generator using the positivity of the energy established in [25]. Our main result is the dispersion decay in the weighted Sobolev norms for solutions with initial states,from the space of continuous spectrum of the Hamilton generator. We also prove the absence of singular spectrum and limiting absorption principle. The multiplicity of every eigenvalue is shown to be infinite. The proofs rely on novel exact bounds and compactness for the inversion of the Bloch generators and on uniform asymptotics for the dispersion relations. We derive the bounds by the energy positivity from [25]. We also use the theory of analytic sets. (C) 2018 Elsevier Inc. All rights reserved.

【 授权许可】

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