By Vincent Rivasseau (Chief Editor)
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Stockmeyer Ann. Henri Poincar´e However, N (ψ, V ψ) ≥ ψ, [− i=1 Zα Zα − ]ψ . |xi − Rˆ n| |xi + Rˆ n| (14) Using (13) and (14) we get ∂E(R) Z 2α 1 ≤− 2 − ∂R 2R 2R N ψ, [Ki − i=1 2Zα 2Zα + Ki − ]ψ . |xi − Rˆ n| |xi + Rˆ n| (15) At this point we note that the lowest eigenvalues of the hydrogenic Herbst operator in each angular momentum channel are bounded from below by the corresponding one of the Klein-Gordon operator (Martin and Roy ). The latter are known explicitly. Since Ki − |xi2Zα −Rˆ n| is just the Herbst operator for a hydrogenic atom (Herbst ) of charge 2Z located at R (location does not matter any more), we have, using the Rayleigh-Ritz principle and following Martin and Roy that −(ψ, Ki − 2Zα ψ) ≤ 1 − |xi − Rˆ n| Finally, from (15) and (16) 1 1+ .
2 (Comparison with the conﬁned Hartree theory). 16) where MH Econf (λ, β) = inf Tr[Π0 Γ]≤λ EβMH [Π0 ΓΠ0 ]. 16) is valid for all values of β > 0 and λ > 0. 3. 15) can be used to estimate the part of the Hartree state ΓH not conﬁned to the lowest Landau band. (1 + λ) εE MH (λ, β) . (1 + λ)3 β −1/2 . 19) 60 B. Baumgartner, R. Seiringer Ann. 3 Conﬁnement for the mean ﬁeld Hamiltonian An analogous result as in the previous subsection holds also for the linearized theory. 4 (Conﬁnement for the mean ﬁeld Hamiltonian).
Olver. Bessel functions of integer order. In Milton Abramowitz and Irene A. Stegun, editors, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, chapter 9, pages 355–433. Dover Publications, New York, 5 edition, 1968.  Mary Beth Ruskai. Limits on Stability of Positive Molecular Ions. Lett. Math. , 18:121–132, 1989.  Mary Beth Ruskai. Absence of bound states in extremely asymmetric positive diatomic molecules. Comm. Math. , 137(3):553–566, 1991.  Jan Philip Solovej.
Annales Henri Poincaré - Volume 2 by Vincent Rivasseau (Chief Editor)
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