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In Region 2, where B ∼ Z 2 , the magnetic and the Coulombic eﬀects are therefore of the same order of magnitude. In Region 1, where B Z 2, the Coulomb eﬀects dominate in all directions. Magnetic eﬀects will not contribute in leading order. In Region 3, where B Z 2 , the magnetic eﬀects dominate the dynamics perpendicular to the magnetic ﬁeld. The electron with low energy is conﬁned to the lowest Landau band, the typical wave functions are squeezed to needles with diameter ∼ B −1/2 . ) Turning from the one-body system to the N -body problem, we remark that Bose statistics has no effect on the size of the ground states, if the pair interactions are ignored.
The proof is immediate from the observation that limx→0 f (x) = 1 and that f (x) < 0 for positive x. 2. Let g(u, v) := K1 (R|u − v|)S(u, v)/|u − v|. The function g fulﬁlls g(u, v) = g(−u, −v). Changing the variable in the integral v := −s yields the desired result. We will now turn to the main goal of this section, namely proving that the function ϕL has its maximum at zero. We will start with the massless case and extend the result to the general massive case. In the latter it is enough for us to assume R ≥ 1.
The asymptotics of the atomic structure and of its energy for large N in the three regions is modeled by energy functionals in generalized Hartree theory. Vol. 2, 2001 Atoms with Bosonic “Electrons” in Strong Magnetic Fields 43 In Region 1 standard Hartree theory is applicable. The energy functional, a functional of the density ρ, is E H [ρ] = 1 ρ(x)d3 x + D[ρ, ρ], |x| |∇ρ1/2 (x)|2 d3 x − where D[ρ, ρ] = 1 2 ρ(x)ρ(y) 3 3 d xd y. 5) Its ground state energy is E H (λ) = inf ρ, ρ=λ E H [ρ]. 6) It is known, [BL83], that lim N→∞ 1 1 E(N, λ, 0) = E H (λ).