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

David Gontier (1, 2) , Mathieu Lewin (1) , Faizan Q. Nazar (1)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 PSL - Université Paris Sciences et Lettres

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.

Domaines: Mathématiques [math]

David Gontier, Mathieu Lewin, Faizan Q. Nazar. The Nonlinear Schrödinger Equation for Orthonormal Functions : Existence of Ground States. Archive for Rational Mechanics and Analysis, 2021, 240 (3), pp.1203-1254. ⟨10.1007/s00205-021-01634-7⟩. ⟨hal-03864212⟩