Annales Henri Poincaré - Volume 2 by Vincent Rivasseau (Chief Editor) PDF

By Vincent Rivasseau (Chief Editor)

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2. (1 + λ)2 β −1/2 )−1 Econf (λ, β). 42) Also this confined Hartree theory has rotation invariant minimizers, since the lowest Landau band is mapped onto itself by rotations around the z-axis. Moreover, since the potential is superharmonic, also the minimizer of this confined theory has L = 0. 5. 24), with general ψ ∈ H1 (R), normalized to ψ 22 = λ: MH (λ, β) = Econf inf Λ, Tr[Λ]=λ, Λ=|χ χ| EβMH [Λ]. 40) is now used as a lower bound 1 C (1 + λ)(λ + Kψ ). 44) The right side of this inequality can be considered as a functional similar to the HSfunctional, but with the constant 1 − C(1 + λ)/L(β) in front of the kinetic energy.

Henri Poincar´e It is clear that Υ (0) = 0. Moreover, Υ (u) ≤ 0 for u ∈ (0, 1) since the function between [ ] in the last equation is convex in (0, 1) and vanishes at u = 0 and u = 1. Hence the maximum of Υ on [0, 1) occurs at 0. To extend the same result for any u we consider the extension of the second statement of Proposition 1 for the massless case and observe that for u ≥ 1 Υ(u) = 1 2 −1 −∞ dv + (u − v)2 2 1 dv −1 sin( π8 (v − 1)) (u − v)2 is a decreasing function of u. This together with the continuity of I proves the lemma.

For Region 2 we introduce a general magnetic Hartree functional, which is studied in detail. It is shown that in the special case of an atom it can be restricted to the subspace of zero angular momentum parallel to the magnetic field, which simplifies the theory considerably. The functional reproduces the energy and the one-particle reduced density matrix for the full N -particle ground state to leading order in N , and it implies the description of the other regions as limiting cases. 1 Introduction The ground states of atoms with many electrons in magnetic fields have been studied in [LSY94a, LSY94b, BSY00], and their energies have been evaluated, exactly to leading order, as some of the physical parameters tend to infinity.

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Annales Henri Poincaré - Volume 2 by Vincent Rivasseau (Chief Editor)


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