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Thus, for the minima of V (x), we have φk < φ < (k + 1)π, θk < θ < ∞, a − b > x2k (θ) > 0, k = 0, 1, . . 39) and for the maxima kπ < φ < φk , ∞ > θ > θk , 0 < x2k+1 (θ) < a − b, k = 1, . . 40) √ Here θk = φk /| sin φk | a − b is the value of θ for which the k’th branch comes into existence. Hence in the interval θK < θ < θK+1 there are exactly 2K + 1 branches, x0, x1 , x2, . . , x2K−1 √ , x2K , forming the K + 1 minima and K maxima of V (x). For 0 < θ < θ0 = 1/ a − b there are no extrema. This classification allows us to discuss the different parameter regimes that arise in the limit of N → ∞.

These operators form a representation of the Lie algebra of SU(1,1) [L3 , L± ] = ±L± . 4) where r = nb /(1 + nb ). e. λn = n for n = 0, 1, . . independent of the nb . Since M = 1 for τ = 0 this is a limiting case for the correlation lengths γξn = 1/λn = 1/n for θ = 0. From Eq. 8) we obtain κn = 1/(1 + n/N ) in the non-interacting case. Hence in the discrete case Rξn = −1/ log κn N/n for N n and this agrees with the values in Figure 9 for n = 1, 2, 3 near τ = 0. 2 Thermal Phase: 0 ≤ θ < 1 In this phase the natural variable is n, not x = n/N .

164]–[166]). 17) describes the pumping of a lossless cavity with a beam of atoms. After k atoms have passed through the cavity, its state has become M k p. In order to see whether this process may reach statistical equilibrium for k → ∞ we write Eq. 20) where Jn = −aqn pn−1 + bqn pn . In statistical equilibrium we must have Jn+1 = Jn , and the common value J = Jn for all n can only be zero since pn , and therefore J , has to vanish for n → ∞. e. 5. There must thus be fewer than 50% excited atoms in the beam, otherwise the lossless cavity blows up.

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