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Simple and generalized condensation in many-boson systems. (English) Zbl 0118.23902

##### Keywords:
mechanics of particles and systems
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 [1] Girardeau, J. Math. Phys. 1 pp 516– (1960) [2] Girardeau, Phys. Rev. 113 pp 755– (1959) [3] Bogoliubov, J. Phys. (U.S.S.R.) 11 pp 23– (1947) [4] Lee, Phys. Rev. 106 pp 1135– (1957) [5] Brueckner, Phys. Rev. 106 pp 1117– (1957) [6] Girardeau, Phys. Rev. 115 pp 1090– (1959) [7] See footnote 11 of I. [8] The expansion (8) is, of caurse, only valid for sufficiently small z. [9] Edwards, Proc. Phys. Soc. (London) 77 pp 433– (1961) [10] To be completely rigorous the replacement should also be carried out in thenandxoccurring in (18), but because of the last inequality (19) this makes no difference in the asymptotic behavior of expectation values. [11] The factor [(n+$$\nu$$)!n!] arises from the replacement nn+$$\nu$$) in the factorn! in (17), the same replacement in the exponent of x-2n-1 gives the factor x-2$$\nu$$, the change $$\delta$$x in thexoccurring in x-2n-1 gives the factor e-2$$\alpha$$ when one makes use of the formula limn(1+x-1$$\delta$$x)-2n-1 = exp(-2limnnx-1$$\delta$$x), the same change in thexoccurring in exp (x2) gives the factor e2$$\beta$$, and the change in thexoccurring inside + gives the third exponential factor. [12] For example, if x = O(1) then $$\delta$$x = O(n-1), and by (21) and (22) $$\alpha$$ = $$\nu$$$$\pi$$3$$\rho$$ + x2$$\psi$$k 1-x4$$\psi$$k2 2d3k-1, $$\beta$$ = 0. If x = O(n 1 2 ) then both $$\alpha$$ and $$\beta$$ are nonvanishing. By (19),xcannot increase more rapidly than O(n 1 2 ) as n. [13] Since the integrals in (28) and (29) run over all ofkspace instead of half of it, it is convenient at this point to modify our notation, taking k to be an even function ofkdefined over all ofkspace. [14] Here and henceforth limn means a limit carried out in accordance with (12). [15] Girardeau, Phys. Rev. 127 pp 1809– (1962) [16] This conclusion could only be made rigorous by showing not only that a solution of (37) exists for the given repulsive interaction, but also that no solution without simple condensation [a solution of (41)] exists, or if a solution of (41) does exist that its energy is higher. Because of the nonlinearity of the variational integral equation, we are not able to supply such proofs; nevertheless, we believe the conclusion to be correct. [17] It is well known that a Bose system whose interaction potential has a Fourier transform which is everywhere finite and satisfies (42) is pathological in that it spontaneously collapses to essentially infinite density. The results of our analysis are therefore only meaningful for such a system when it is understood that they refer to a system with a given,fixed, finite, spacially constant density. Actually this is only a necessary condition for meaningful results, but not a sufficient one; it is also necessary to require that the two-particle distribution function D(r,r) approaches a positive constant (unity whenDis properly normalized) for r-r0 where $$\Omega$$-13r0 as $$\Omega$$, in order to avoid, e.g., pathological states (”droplets”) for which D(r,r) vanishes when r-r0. These pathological properties cannot arise, and hence the corresponding restrictions need not be applied, if the attractive interaction includes a hard core; but then the interaction $$\nu$$(k) in momentum space does not exist. The difficulties reappear if one treats the hard core in a lowest-order pseudopotential (scattering length) approximation, in which case the aforementioned two restrictions must be applied; they are in fact automatically satisfied by the variational ansatz (1). [18] We assume that $$\nu$$(k) does not differ appreciably from $$\nu$$(0) forkmuch less than some finite range a-1. [19] It had previously been shown that one could already obtain an energy twice as low as that of the unperturbed state by using a free-particle trial state (simultaneous eigen-state of the Nk) which allows for ”momentum smearing”; see M. Girardeau (Boeing Document D1-82-0119). The result (47) shows that the off-diagonal terms in the interaction contribute an additional - 1 2 $$\rho$$$$\nu$$(0) when one uses a pair-type state corresponding to (43). [20] Since the expression (39) is only correct toO(n), its invariance under the transformation (50) likewise only holds toO(n). [21] One has then Jgr;k = -1+O([f(n)]-1), ks. [22] Equation (59) can also be obtained directly from (37) by assuming k to be real, dropping the terms proportional to f0 [because of (38)], and substituting (55). [23] The kinetic energy contribution from the singular part of k vanishes in the limit (12) because of the inequalities (2$$\pi$$)-3$$\rho$$-1 limko limn k
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