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

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SYSNO ASEP 0462436 Document Type J - Journal Article R&D Document Type **Journal Article**Subsidiary J Článek ve WOS Title The Hardy inequality and the heat equation with magnetic field in any dimension Author(s) **Cazacu, C. (RO)**_{}__Krejčiřík, David__(UJF-V)_{ RID}Number of authors 2 Source Title Communications in Partial Differential Equations - ISSN 0360-5302

Roč. 41, č. 7 (2016), s. 1056-1088Number of pages 33 s. Publication form Print - P Language eng - English Country US - United States Keywords Aharonov-Bohm magnetic field ; Hardy inequality ; heat equation ; large time behaviour of solutions ; magnetic Schrodinger operator Subject RIV BE - Theoretical Physics R&D Projects GA14-06818S GA ČR - Czech Science Foundation (CSF) Institutional support UJF-V - RVO:61389005 UT WOS 000380142200003 EID SCOPUS 84975282623 DOI 10.1080/03605302.2016.1179317 Annotation 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. Workplace Nuclear Physics Institute Contact Markéta Sommerová, sommerova@ujf.cas.cz, Tel.: 266 173 228 ; Renata Glasová, glasova@ujf.cas.cz, Tel.: 266 177 223 Year of Publishing 2017

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