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By Elliott H. Lieb, Robert Seiringer, Jan Philip Solovej, Jakob Yngvason
This booklet encompasses a targeted survey of the mathematically rigorous effects in regards to the quantum-mechanical many-body challenge which were got through the authors long ago seven years. It addresses an issue that isn't in basic terms wealthy mathematically, utilizing a wide number of recommendations in mathematical research, yet can be one with powerful ties to present experiments on ultra-cold Bose gases and Bose-Einstein condensation. The publication presents a pedagogical access into an energetic sector of ongoing learn for either graduate scholars and researchers. it truly is an outgrowth of a path given by way of the authors for graduate scholars and post-doctoral researchers on the Oberwolfach learn Institute in 2004. The e-book additionally offers a coherent precis of the sector and a reference for mathematicians and physicists lively in learn on quantum mechanics.
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Chapter 3 The Dilute Bose Gas in 2D In contrast to the three-dimensional theory, the two-dimensional Bose gas began to receive attention only relatively late. The first derivation of the correct asymptotic formula was, to our knowledge, done by Schick [S] for a gas of hard discs. He found e(ρ) ≈ 4πµρ| ln(ρa2 )|−1 . 1) This was accomplished by an infinite summation of ‘perturbation series’ diagrams. Subsequently, a corrected modification of [S] was given in [HFM]. Positive temperature extensions were given in [Po] and in [FH].
Finally, one lets the box size tend to zero. However, it is not possible to simply approximate V by a constant potential in each box. , v = 0 and hence a = 0. Here E0 = N ω, but a ‘naive’ box method gives only minx V (x) as lower bound, since it clearly pays to put all the particles with a constant wave function in the box with the lowest value of V . For this reason we start by separating out the GP wave function in each variable and write a general wave function Ψ as N φGP (xi )F (x1 , . . , xN ).
9) is essentially a consequence of the weak lower semicontinuity of the L2 norm, but the dependence on Ωn leads to a slight complication. First, Eq. 8) and Cn → ∞ clearly imply that |Ωcn | → 0, because ∇ϕ hn 2L2 (K) > ϕ2 . By choosing a subsequence we may assume that n |Ωcn | < ∞. For some fixed ˜ N ⊂ Ωn for n ≥ N . Since ∇ϕ hn 2 2 N let ΩN = K \ ∪n≥N Ωcn . Then Ω L (Ωn ) is bounded, ∇ϕ hn is also bounded in L2 (ΩN ) and a subsequence of it converges weakly in L2 (ΩN ) to ∇ϕ h. Hence lim inf ∇ϕ hn n→∞ 2 L2 (Ωn ) ≥ lim inf ∇ϕ hn n→∞ 2 L2 (ΩN ) ≥ ∇ϕ h 2 L2 (ΩN ) .