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Sharp upper bounds for the density of relativistic atoms: Noninteracting case

Rupert L. Frank, Konstantin Merz

Abstract

We prove an optimal upper bound for the density of electrons of an infinite Bohr atom (no electron-electron interactions) described by the relativistic operators of Chandrasekhar and Dirac. We also consider densities in each angular momentum channel separately.

Sharp upper bounds for the density of relativistic atoms: Noninteracting case

Abstract

We prove an optimal upper bound for the density of electrons of an infinite Bohr atom (no electron-electron interactions) described by the relativistic operators of Chandrasekhar and Dirac. We also consider densities in each angular momentum channel separately.

Paper Structure

This paper contains 14 sections, 8 theorems, 170 equations.

Table of Contents

  1. Introduction and main results
  2. The Chandrasekhar--Coulomb operator
  3. The Dirac--Coulomb operator
  4. Background: the strong Scott conjecture
  5. Optimal Small-distance bounds
  6. A generalization
  7. Proof of Theorem \ref{['boundrho']}
  8. Electrons with prescribed angular momentum
  9. Optimal small-distance bounds at fixed angular momentum
  10. Proof of Theorem \ref{['main']}
  11. Dirac--Coulomb operator
  12. Reduction to bounds for fixed angular momentum
  13. Bounds for fixed angular momentum. I
  14. Bounds for fixed angular momentum. II

Key Result

Theorem 1.1

Let $0<\kappa\leq 2/\pi$. Then there is a constant $A_\kappa<\infty$ such that for all $x\in\mathbb{R}^3$, Here $\eta_\kappa$ is the unique number in $(0,1]$ such that $(1-\eta_\kappa)\tan\frac{\pi\eta_\kappa}{2} = \kappa$. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (20)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Remark 1.4
  • Theorem 2.1
  • Remark 2.2
  • Theorem 3.1
  • Remark 3.2
  • proof : Proof of Theorem \ref{['boundrhoell']}
  • Theorem 3.3
  • ...and 10 more