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

2020/02/29 by David Gontier, Mathieu Lewin, Faizan Q. Nazar · 1 citation
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #Complex system #Dirac (video compression format) #Exponent #Infinity #Nonlinear Photonic Systems #Nonlinear system #Orthonormal basis #Sequence (biology) #Spectral Theory in Mathematical Physics #Symmetry (geometry) #Translational symmetry #math-ph #math.AP #math.MP #math.SP

paper · pdf · doi:10.1007/s00205-021-01634-7

Final version, to appear in Arch. Rat. Mech. Anal

openalex created_date 2020/02/24 · arxiv created 2021/03/17 · openalex publication_date 2021/04/20 · arxiv updated 2021/05/05 · openalex updated_date 2026/08/05

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.

Citations

• Bound for the Kinetic Energy of Fermions Which Proves the Stability of Matter
• Uniqueness of positive solutions of Δu−u+up=0 in Rn
• Accuracy of mean field approximations for atoms and molecules
• Non-perturbative embedding of local defects in crystalline materials
• Nonlinear Schr�dinger equations and sharp interpolation estimates
• Generalized Hartree-Fock theory and the Hubbard model
• Nonlinear scalar field equations, I existence of a ground state
• Stability of periodic waves of finite amplitude on the surface of a deep fluid
• Analysis

Cited by

• The Nonlinear Schrödinger Equation for Orthonormal Functions II: Application to Lieb–Thirring Inequalities