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2ψ̃†+2Φψ ̃†ψ̃+Φ*ψ̃ψ̃+ ψ̃†ψ̃ψ̃ and ψφ†φ = Φφ†φ+ψ̃φ†φ. From the semiclassical point of view, the cubic product of the operator, ψ̃†ψ̃ψ̃ (or ψ̃φ†φ), accounts for the collisions involving the condensate and noncondensate atoms (or fermions) [18]. In the collisionless regime we may assume that these products have only a negligible effect on the dynamics of the condensate and we may safely set the triplet average value to zero: <ψ̃†(r,t)ψ̃(r,t)ψ̃(r,t)> = 0 and <ψ̃(r,t)φ†(r,t)φ(r,t)> = 0. On the other hand, for a pure Bose gas the damping due to collisions has been discussed by Williams and Griffin in Ref. [21]. For typical trap parameters, it is found to be 2 or 3 times smaller than the Landau damping arising from the dynamical mean-field effect as discussed in the present paper. The collisional frequency shift is also considered in these papers, and found to vanish in the first-order perturbation treatment.
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0242460157
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0(r) also becomes negligible. While at high temperatures, it approaches the finite-temperature Hartree-Fock spectrum. Therefore it is expected to give a reasonable first approximation for the excitation spectrum in Bose gases at all temperatures [14].
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37
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85088492379
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note
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f(r) are an order smaller than δΦ(r).
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39
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85088489612
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note
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F + ω.
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0(r+x/2,r-x/2; ω = 0) that diverges as 1/x for x → 0 [30].
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