Hardy–Lieb–Thirring Inequalities for Fractional Pauli Operators

TY - JOUR

T1 - Hardy–Lieb–Thirring Inequalities for Fractional Pauli Operators

AU - Bley, Gonzalo A.

AU - Fournais, Søren

PY - 2019

Y1 - 2019

N2 - We provide lower bounds for the sum of the negative eigenvalues of the operator | σ· pA| 2 s- Cs/ | 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, pA= p- A, with p= - i∇ the three-dimensional momentum operator and A a given magnetic vector potential, and Cs is the critical Hardy constant, that is, the optimal constant in the Hardy inequality | p| 2 s≥ Cs/ | 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∫V-1+3/(2s), for a positive constant Ms. 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‖22 and integrals of powers of the negative part of V.

AB - We provide lower bounds for the sum of the negative eigenvalues of the operator | σ· pA| 2 s- Cs/ | 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, pA= p- A, with p= - i∇ the three-dimensional momentum operator and A a given magnetic vector potential, and Cs is the critical Hardy constant, that is, the optimal constant in the Hardy inequality | p| 2 s≥ Cs/ | 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∫V-1+3/(2s), for a positive constant Ms. 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‖22 and integrals of powers of the negative part of V.

KW - ASYMPTOTICS

KW - BOUND-STATES

KW - COULOMB-SYSTEMS

KW - FERMIONS

KW - HEAVY-ATOMS

KW - INSTABILITY

KW - NONHOMOGENEOUS MAGNETIC-FIELDS

KW - RELATIVISTIC MATTER

KW - SIMPLE PROOF

KW - STABILITY

UR - http://www.scopus.com/inward/record.url?scp=85051267537&partnerID=8YFLogxK

U2 - 10.1007/s00220-018-3204-y

DO - 10.1007/s00220-018-3204-y

M3 - Journal article

AN - SCOPUS:85051267537

SN - 0010-3616

VL - 365

SP - 651

EP - 683

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

IS - 2

ER -