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Abstract
We provide lower bounds for the sum of the negative eigenvalues of the operator  σ· p_{A} ^{2} ^{s} C_{s}/  x ^{2} ^{s}+ V in three dimensions, where s∈ (0 , 1] , covering the interesting physical cases s = 1 and s = 1/2. Here σ is the vector of Pauli matrices, p_{A}= p A, with p=  i∇ the threedimensional momentum operator and A a given magnetic vector potential, and C_{s} is the critical Hardy constant, that is, the optimal constant in the Hardy inequality  p ^{2} ^{s}≥ C_{s}/  x ^{2} ^{s}. If spin is neglected, results of this type are known in the literature as Hardy–Lieb–Thirring inequalities, which bound the sum of negative eigenvalues from below by Ms∫V1+3/(2s), for a positive constant M_{s}. The inclusion of magnetic fields in this case follows from the nonmagnetic case by diamagnetism. The addition of spin, however, offers extra challenges that make the result more elusive. It is the purpose of this article to resolve this problem by providing simple bounds for the sum of the negative eigenvalues of the operator in question. In particular, for 1 / 2 ≤ s≤ 1 we are able to express the bound purely in terms of the magnetic field energy ‖B‖22 and integrals of powers of the negative part of V.
Original language  English 

Journal  Communications in Mathematical Physics 
Volume  365 
Issue  2 
Pages (fromto)  651683 
Number of pages  33 
ISSN  00103616 
DOIs  
Publication status  Published  2019 
Keywords
 ASYMPTOTICS
 BOUNDSTATES
 COULOMBSYSTEMS
 FERMIONS
 HEAVYATOMS
 INSTABILITY
 NONHOMOGENEOUS MAGNETICFIELDS
 RELATIVISTIC MATTER
 SIMPLE PROOF
 STABILITY
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Dive into the research topics of 'Hardy–Lieb–Thirring Inequalities for Fractional Pauli Operators'. Together they form a unique fingerprint.Projects
 1 Finished

Semiclassical Quantum Mechanics
Fournais, S. (PI), Madsen, P. (Participant), Mikkelsen, S. (Participant), Miqueu, J.P. C. (Participant) & Bley, G. (Participant)
01/07/2015 → 31/12/2020
Project: Research