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Extra info for Annales Henri Poincaré - Volume 2
12 (Uniqueness of ΓH ). The functional EβMH has a unique minimizer ΓH under the condition Tr[Γ] ≤ λ, which is proportional to the projection onto the positive function ρH . In particular, ΓH has rank 1. 11) under the condition ρ ≤ λ. It is C ∞ away from the origin, continuous at x = 0, and strictly positive. 66) where µ is the chemical potential of the MH theory. 12 proves the assertion made in the Introduction that both E MH and EˆMH have the same ground state energy and density. 1 (Hartree energy for small β).
46) Especially, for all δ > 0, MH MH EβMH [(1 − δ)ΓH + δΓ] = (1 − δ)Eβ,lin [ΓH ] + δEβ,lin [Γ] − D[ρH , ρH ] +δ 2 D[ρΓ − ρH , ρΓ − ρH ]. 47) MH MH Now if there exists a Γ0 with Tr[Γ0 ] ≤ λ and Eβ,lin [Γ0 ] < Eβ,lin [ΓH ] we can choose δ small enough to conclude that MH EβMH [(1 − δ)ΓH + δΓ0 ] < Eβ,lin [ΓH ] − D[ρH , ρH ] = EβMH [ΓH ], which contradicts the fact that ΓH minimizes EβMH . 48) 54 B. Baumgartner, R. Seiringer Ann. Henri Poincar´e The Hartree equation, the Euler-Lagrange equation corresponding to the minimization of the functional EβMH , is H H Γ = µΓ.
N ≤ N0 , it is suﬃcient to use the 1/R2 estimate which is the nonrelativistic estimate. However, for large N , namely N ≥ N0 , the 1/R estimate, which is the ultrarelativistic estimate, yields the better results. 042, assuming, of course, that Zα < 1/4. A To prove the equation (25) we ﬁrst introduce F (t) := 1 R cos(ψ(t1 ))2 (t1 − 1)2 + t2 2 + t3 2 sin(ψ(t1 ))2 + (t1 + 1)2 + t2 2 + t3 2 . (56) For t1 ≤ −1 we ﬁnd that F (t) = 1 R 1 (t1 − 1)2 2 + t2 + t3 2 ≤ 1 1 ≤ . |t1 − 1| R (57) Similarly we get F (t) ≤ 1/R for t1 ≥ 1.