The Nonlinear Schrödinger Equation for Orthonormal Functions: Existence of Ground States

David Gontier; Mathieu Lewin; Faizan Q. Nazar

doi:10.1007/s00205-021-01634-7

The Nonlinear Schrödinger Equation for Orthonormal Functions: Existence of Ground States

David Gontier
, Mathieu Lewin
, Faizan Q. Nazar

Université Paris Dauphine

Résultats de recherche: Contribution à un journal › Article › Revue par des pairs

Résumé

We study the nonlinear Schrödinger equation for systems of N orthonormal functions. We prove the existence of ground states for all N when the exponent p of the non linearity is not too large, and for an infinite sequence N_j tending to infinity in the whole range of possible p’s, in dimensions d≥ 1. This allows us to prove that translational symmetry is broken for a quantum crystal in the Kohn–Sham model with a large Dirac exchange constant.

langue originale	Anglais
Pages (de - à)	1203-1254
Nombre de pages	52
journal	Archive for Rational Mechanics and Analysis
Volume	240
Numéro de publication	3
Les DOIs	https://doi.org/10.1007/s00205-021-01634-7
état	Publié - 1 juin 2021
Modification externe	Oui

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10.1007/s00205-021-01634-7

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abstract = "We study the nonlinear Schr{\"o}dinger equation for systems of N orthonormal functions. We prove the existence of ground states for all N when the exponent p of the non linearity is not too large, and for an infinite sequence Nj tending to infinity in the whole range of possible p{\textquoteright}s, in dimensions d≥ 1. This allows us to prove that translational symmetry is broken for a quantum crystal in the Kohn–Sham model with a large Dirac exchange constant.",

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AB - We study the nonlinear Schrödinger equation for systems of N orthonormal functions. We prove the existence of ground states for all N when the exponent p of the non linearity is not too large, and for an infinite sequence Nj tending to infinity in the whole range of possible p’s, in dimensions d≥ 1. This allows us to prove that translational symmetry is broken for a quantum crystal in the Kohn–Sham model with a large Dirac exchange constant.

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The Nonlinear Schrödinger Equation for Orthonormal Functions: Existence of Ground States

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