• xmlui.mirage2.page-structure.header.title
    • français
    • English
  • Help
  • Login
  • Language 
    • Français
    • English
View Item 
  •   BIRD Home
  • CEREMADE (UMR CNRS 7534)
  • CEREMADE : Publications
  • View Item
  •   BIRD Home
  • CEREMADE (UMR CNRS 7534)
  • CEREMADE : Publications
  • View Item
JavaScript is disabled for your browser. Some features of this site may not work without it.

Browse

BIRDResearch centres & CollectionsBy Issue DateAuthorsTitlesTypeThis CollectionBy Issue DateAuthorsTitlesType

My Account

LoginRegister

Statistics

Most Popular ItemsStatistics by CountryMost Popular Authors
Thumbnail

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

Esteban, Maria J.; Lewin, Mathieu; Séré, Eric (2021), Dirac-Coulomb operators with general charge distribution. II. The lowest eigenvalue, Proceedings of the London Mathematical Society, 123, 4, p. 345-383. 10.1112/plms.12396

View/Open
2003.04051.pdf (621.9Kb)
Type
Article accepté pour publication ou publié
Date
2021
Journal name
Proceedings of the London Mathematical Society
Volume
123
Number
4
Publisher
Wiley
Pages
345-383
Publication identifier
10.1112/plms.12396
Metadata
Show full item record
Author(s)
Esteban, Maria J. cc
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Lewin, Mathieu cc
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Séré, Eric
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Abstract (EN)
Consider the Coulomb potential −μ∗|x|−1 generated by a non-negative finite measure μ. It is well known that the lowest eigenvalue of the corresponding Schrödinger operator −Δ/2−μ∗|x|−1 is minimized, at fixed mass μ(R3)=ν, when μ 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. In a previous work on the subject we proved that this operator is self-adjoint when μ has no atom of mass larger than or equal to 1, and that its eigenvalues are given by min-max formulas. Here we consider the critical mass ν1, below which the lowest eigenvalue does not dive into the lower continuum spectrum for all μ≥0 with μ(R3)<ν1. We first show that ν1 is related to the best constant in a new scaling-invariant Hardy-type inequality. Our main result is that for all 0≤ν<ν1, there exists an optimal measure μ≥0 giving the lowest possible eigenvalue at fixed mass μ(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. The existence proof is based on the concentration-compactness principle.
Subjects / Keywords
Dirac-Coulomb operators; min-max formulas; variational methods; eigenvalue optimization; spectral theory; concentration-compactness principle; unique continuation principle

Related items

Showing items related by title and author.

  • Thumbnail
    Dirac-Coulomb operators with general charge distribution I. Distinguished extension and min-max formulas 
    Esteban, Maria J.; Lewin, Mathieu; Séré, Eric (2021) Article accepté pour publication ou publié
  • Thumbnail
    Distinguished self-adjoint extension and eigenvalues of operators with gaps. Application to Dirac-Coulomb operators 
    Dolbeault, Jean; Esteban, Maria J.; Séré, Eric (2022) Document de travail / Working paper
  • Thumbnail
    Domains for Dirac-Coulomb min-max levels 
    Esteban, Maria J.; Lewin, Mathieu; Séré, Eric (2019) Article accepté pour publication ou publié
  • Thumbnail
    General results on the eigenvalues of operators with gaps, arising from both ends of the gaps. Application to Dirac operators. 
    Dolbeault, Jean; Esteban, Maria J.; Séré, Eric (2006) Article accepté pour publication ou publié
  • Thumbnail
    On the Eigenvalues of Operators with Gaps. Application to Dirac Operators 
    Dolbeault, Jean; Esteban, Maria J.; Séré, Eric (2000) Article accepté pour publication ou publié
Dauphine PSL Bibliothèque logo
Place du Maréchal de Lattre de Tassigny 75775 Paris Cedex 16
Phone: 01 44 05 40 94
Contact
Dauphine PSL logoEQUIS logoCreative Commons logo