Hardy–Lieb–Thirring Inequalities for Fractional Pauli Operators

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 three...

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Published in:Communications in mathematical physics Vol. 365; no. 2; pp. 651 - 683
Main Authors: Bley, Gonzalo A., Fournais, Søren
Format: Journal Article
Language:English
Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.01.2019
Springer Nature B.V
Subjects:
Classical and Quantum Gravitation
Complex Systems
Diamagnetism
Eigenvalues
Inequalities
Lower bounds
Magnetic fields
Magnetic vector potentials
Mathematical analysis
Mathematical and Computational Physics
Mathematical Physics
Matrix algebra
Matrix methods
Operators (mathematics)
Physics
Physics and Astronomy
Quantum Physics
Relativity Theory
Theoretical
ISSN:0010-3616, 1432-0916
Online Access:Get full text
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Summary: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 three-dimensional 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 - M s ∫ V - 1 + 3 / ( 2 s ) , for a positive constant M s . The inclusion of magnetic fields in this case follows from the non-magnetic 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 ‖ 2 2 and integrals of powers of the negative part of V .
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ISSN:0010-3616
1432-0916
DOI:10.1007/s00220-018-3204-y