Journal of Statistical Physics

, Volume 116, Issue 1–4, pp 475–506 | Cite as

Magnetic Lieb—Thirring Inequalities with Optimal Dependence on the Field Strength

  • László Erdős
  • Jan Philip Solovej

Abstract

The Pauli operator describes the energy of a nonrelativistic quantum particle with spin 1/2 in a magnetic field and an external potential. Bounds on the sum of the negative eigenvalues are called magnetic Lieb–Thirring (MLT) inequalities. The purpose of this paper is twofold. First, we prove a new MLT inequality in a simple way. Second, we give a short summary of our recent proof of a more refined MLT inequality(8) and we explain the differences between the two results and methods. The main feature of both estimates, compared to earlier results, is that in the large field regime they grow with the optimal (first) power of the strength of the magnetic field. As a byproduct of the method, we also obtain optimal upper bounds on the pointwise density of zero energy eigenfunctions of the Dirac operator.

kernel of Dirac operator non-homogeneous magnetic field 

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Copyright information

© Plenum Publishing Corporation 2004

Authors and Affiliations

  • László Erdős
    • 1
    • 2
  • Jan Philip Solovej
    • 3
  1. 1.School of MathematicsGeorgia Tech Atlanta
  2. 2.Mathematisches Institut LMUMünichGermany
  3. 3.Department of MathematicsUniversity of CopenhagenCopenhagenDenmark

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