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

Research output: Contribution to journal › Article › peer-review

Abstract

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.

Original language	English
Pages (from-to)	1203-1254
Number of pages	52
Journal	Archive for Rational Mechanics and Analysis
Volume	240
Issue number	3
DOIs	https://doi.org/10.1007/s00205-021-01634-7
Publication status	Published - 1 Jun 2021
Externally published	Yes

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title = "The Nonlinear Schr{\"o}dinger Equation for Orthonormal Functions: Existence of Ground States",

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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N2 - 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.

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

Abstract

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