Thumbnail

Keller estimates of the eigenvalues in the gap of Dirac operators

Dolbeault, Jean; Gontier, David; Pizzichillo, Fabio; Van Den Bosch, Hanne (2022), Keller estimates of the eigenvalues in the gap of Dirac operators. https://basepub.dauphine.psl.eu/handle/123456789/23101

Voir/Ouvrir
Type
Document de travail / Working paper
Date
2022
Titre de la collection
Cahiers du CEREMADE
Pages
35
Métadonnées
Afficher la notice complète
Auteur(s)
Dolbeault, Jean cc
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Gontier, David cc
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Pizzichillo, Fabio
Departamento de Matemática Aplicada y Ciencias de la Computación
Van Den Bosch, Hanne
Departamento de Ingeniería Matemática [Santiago] [DIM]
Center for Mathematical Modeling [CMM]
Résumé (EN)
We estimate the lowest eigenvalue in the gap of a Dirac operator with mass in terms of a Lebesgue norm of the potential. Such a bound is the counterpart for Dirac operators of the Keller estimates for the Schrödinger operator, which are equivalent to Gagliardo-Nirenberg-Sobolev interpolation inequalities. Domain, self-adjointness, optimality and critical values of the norms are addressed, while the optimal potential is given by a Dirac equation with a Kerr nonlinearity. A new critical bound appears, which is the smallest value of the norm of the potential for which eigenvalues may reach the bottom of the gap in the essential spectrum. Most of our result are established in the Birman-Schwinger reformulation of the problem.
Mots-clés
Dirac operators; potential; spectral gap; eigenvalues; ground state; min-max principle; Birman-Schwinger operator; domain; self-adjoint operators; Keller estimate; interpolation; Gagliardo-Nirenberg-Sobolev inequality; Kerr nonlinearity