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

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    0462436 - ÚJF 2017 RIV US eng J - Journal Article
    Cazacu, C. - Krejčiřík, David
    The Hardy inequality and the heat equation with magnetic field in any dimension.
    Communications in Partial Differential Equations. Roč. 41, č. 7 (2016), s. 1056-1088. ISSN 0360-5302
    R&D Projects: GA ČR(CZ) GA14-06818S
    Institutional support: RVO:61389005
    Keywords : Aharonov-Bohm magnetic field * Hardy inequality * heat equation * large time behaviour of solutions * magnetic Schrodinger operator
    Subject RIV: BE - Theoretical Physics
    Impact factor: 1.608, year: 2016

    n 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.
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