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The Hardy inequality and the heat equation with magnetic field in any dimension

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    SYSNO ASEP0462436
    Document TypeJ - Journal Article
    R&D Document TypeJournal Article
    Subsidiary JČlánek ve WOS
    TitleThe Hardy inequality and the heat equation with magnetic field in any dimension
    Author(s) Cazacu, C. (RO)
    Krejčiřík, David (UJF-V) RID
    Number of authors2
    Source TitleCommunications in Partial Differential Equations - ISSN 0360-5302
    Roč. 41, č. 7 (2016), s. 1056-1088
    Number of pages33 s.
    Publication formPrint - P
    Languageeng - English
    CountryUS - United States
    KeywordsAharonov-Bohm magnetic field ; Hardy inequality ; heat equation ; large time behaviour of solutions ; magnetic Schrodinger operator
    Subject RIVBE - Theoretical Physics
    R&D ProjectsGA14-06818S GA ČR - Czech Science Foundation (CSF)
    Institutional supportUJF-V - RVO:61389005
    UT WOS000380142200003
    EID SCOPUS84975282623
    Annotationn the Euclidean space of any dimension d, we consider the heat semi group generated by the magnetic Schrodinger operator from which an inverse-square potential is subtracted to make the operator critical in the magnetic-free case. Assuming that the magnetic field is compactly supported, we show that the polynomial large-time behavior of the heat semigroup is determined by the eigenvalue problem for a magnetic Schrodinger operator on the (d-1)-dimensional sphere whose vector potential reflects the behavior of the magnetic field at the space infinity. From the spectral problem on the sphere, we deduce that in d = 2 there is an improvement of the decay rate of the heat semigroup by a polynomial factor with power proportional to the distance of the total magnetic flux to the discrete set of flux quanta, while there is no extra polynomial decay rate in higher dimensions. To prove the results, we establish new magnetic Hardy-type inequalities for the Schrodinger operator and develop the method of self-similar variables and weighted Sobolev spaces for the associated heat equation.
    WorkplaceNuclear Physics Institute
    ContactMarkéta Sommerová,, Tel.: 266 173 228 ; Renata Glasová,, Tel.: 266 177 223
    Year of Publishing2017
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