Kinetic energy estimates for the accuracy of the time-dependent Hartree–Fock approximation with Coulomb interaction

Abstract

We study the time evolution of a system of N spinless fermions in R 3 which interact through a pair potential, e.g., the Coulomb potential. We compare the dynamics given by the solution to Schrödinger's equation with the time-dependent Hartree-Fock approximation, and we give an estimate for the accuracy of this approximation in terms of the kinetic energy of the system. This leads, in turn, to bounds in terms of the initial total energy of the system. MSC class: 35Q40, 35Q55, 81Q05, 82C10

Semi-classical limit of large fermionic systems at positive temperature

“…The one-particle densities d ρ (1) Γ N converge to ν strongly in L 1 R d and are uniformly bounded in L ∞ (R d ). Hence they converge strongly in all…”

“…Furthermore, we can take ρ (1) Γ N to be supported in a compact set which is independent of N and uniformly bounded in L ∞ (R d ) so that the convergence (21) holds in fact in all L p (R d ) for 1 ≤ p < ∞.…”

“…Finally, denoting by Γ z,F =1 the uncorrelated version of Γ z and by ρ (1) z,F =1 its one-body density we have…”

“…We begin with the case η = 0. Using coherent states, the tightness of (m (1) f,Γ N ) N and a finite volume approximation we obtain…”

“…For atoms the positive Thomas-Fermi model was derived for the first time in [43]. There are several mathematical works on the time-dependent setting [42,52,3,14,17,8,5,1,44,7,13], in which the Schrödinger dynamics has been proved to converge to the Vlasov time-dependent equation in the limit N → ∞. Finally, the first two terms in the expansion of the (free) energy of a Fermi gas with spin in the limit ρ → 0 was provided in [32] at T = 0 and in [50] at T > 0.…”