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

Gontier, David; Lewin, Mathieu; Nazar, Faizan Q. (2021), The Nonlinear Schrödinger Equation for Orthonormal Functions : Existence of Ground States, Archive for Rational Mechanics and Analysis, 240, 3, p. 1203-1254. 10.1007/s00205-021-01634-7

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Type
Article accepté pour publication ou publié
Date
2021
Journal name
Archive for Rational Mechanics and Analysis
Volume
240
Number
3
Publisher
Springer
Pages
1203-1254
Publication identifier
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Author(s)
Gontier, David cc
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Lewin, Mathieu cc
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Nazar, Faizan Q.
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Abstract (EN)
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.