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Your search keyword '"81Q10, 49R05, 49J35, 47A75, 47B25"' showing total 2 results
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2 results on '"81Q10, 49R05, 49J35, 47A75, 47B25"'

1. Keller and Lieb-Thirring estimates of the eigenvalues in the gap of Dirac operators

Author

Dolbeault, Jean, Gontier, David, Pizzichillo, Fabio, and Bosch, Hanne Van Den

Subjects
Mathematics - Analysis of PDEs, 81Q10, 49R05, 49J35, 47A75, 47B25
Abstract

We estimate the lowest eigenvalue in the gap of the essential spectrum 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 Schr\"odinger 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. The Keller estimate is then extended to a Lieb-Thirring inequality for the eigenvalues in the gap. Most of our result are established in the Birman-Schwinger reformulation.

Published
2022

2. Keller estimates of the eigenvalues in the gap of Dirac operators

Author

Dolbeault, Jean, Gontier, David, Pizzichillo, Fabio, Bosch, Hanne Van Den, CEntre de REcherches en MAthématiques de la DEcision (CEREMADE), Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS), Analyse (DMA), Département de Mathématiques et Applications - ENS Paris (DMA), École normale supérieure - Paris (ENS-PSL), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-École normale supérieure - Paris (ENS-PSL), Departamento de Matemática Aplicada y Ciencias de la Computación, Universidad de Cantabria, Departamento de Ingeniería Matemática [Santiago] (DIM), Universidad de Chile = University of Chile [Santiago] (UCHILE)-Centre National de la Recherche Scientifique (CNRS), Center for Mathematical Modeling (CMM), Universidad de Chile = University of Chile [Santiago] (UCHILE), ANID/Basal project #FB210005, ANID/Basal project #ACE210010, ANID/Fondecyt project #11220194, MathAmSud project EEQUADDII 20-MATH-04, Project PID2021-123034NB-I00 of the MCIN/AEI/10.13039/501100011033/ FEDER, UE, ANR-17-CE40-0030,EFI,Entropie, flots, inégalités(2017), ANR-17-CE29-0004,molQED,Electrodynamique Quantique Moléculaire(2017), and European Project: 725528,MDFT

Subjects
potential, domain, Keller estimate, Dirac operators, eigenvalues, min-max principle, Kerr nonlinearity, interpolation, spectral gap, self-adjoint operators, 81Q10, 49R05, 49J35, 47A75, 47B25, FOS: Mathematics, Gagliardo-Nirenberg-Sobolev inequality, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], ground state, Birman-Schwinger operator, Analysis of PDEs (math.AP)
Abstract

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.

Published
2022

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