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Dirac-Coulomb operators with general charge distribution. II. The lowest eigenvalue

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2003.04051.pdf (391.9Kb)
Date
2020
Publisher city
Paris
Publisher
Cahier de recherche CEREMADE, Université Paris-Dauphine
Publishing date
03-2020
Link to item file
https://hal.archives-ouvertes.fr/hal-02503462
Dewey
Sciences connexes (physique, astrophysique)
Sujet
Dirac-Coulomb operators
URI
https://basepub.dauphine.fr/handle/123456789/20607
Collections
  • CEREMADE : Publications
Metadata
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Author
Esteban, Maria J.
60 CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Lewin, Mathieu
60 CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Séré, Eric
60 CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Type
Document de travail / Working paper
Item number of pages
39
Abstract (EN)
Consider an electron moving in the Coulomb potential −μ∗|x|−1−μ∗|x|−1 generated by any non-negative finite measure μμ. It is well known that the lowest eigenvalue of the corresponding Schrodinger operator −Δ/2−μ∗|x|−1−Δ/2−μ∗|x|−1 is minimized, at fixed mass μ(R3)=νμ(R3)=ν, when μμ is proportional to a delta. In this paper we investigate the conjecture that the same holds for the Dirac operator −iα⋅∇+β−μ∗|x|−1−iα⋅∇+β−μ∗|x|−1. In a previous work on the subject we proved that this operator is well defined and that its eigenvalues are given by min-max formulas. Here we show that there exists a critical number ν1ν1 below which the lowest eigenvalue does not dive into the lower continuum spectrum, for all μ≥0μ≥0 with μ(R3)<ν1μ(R3)<ν1. Our main result is that for all 0≤ν<ν10≤ν<ν1, there exists an optimal measure μ≥0 μ≥0 giving the lowest possible eigenvalue at fixed mass μ(R3)=νμ(R3)=ν, which concentrates on a compact set of Lebesgue measure zero. The last property is shown using a new unique continuation principle for Dirac operators.

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