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By Ladislav Šamaj
"Including subject matters now not commonly coated within the literature, comparable to (1 + 1)- dimensional quantum box thought and classical two-dimensional Coulomb gases, this publication considers quite a lot of versions and demonstrates a couple of occasions to which they are often applied."--
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Extra resources for Introduction to the statistical physics of integrable many-body systems
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68) where Ij = 0, ±1, ±2, . . , if N odd, if N even. ± 12 , ± 32 , . . 69) Only solutions with distinct wavenumbers are allowed to avoid the nullity of the wavefunction. Owing to θ(k)+θ(−k) = 0, the total particle momentum is given by 2π K = L N Ij. 70) j=1 The wavefunction in the ordering sector Q ∈ S N is formally expressible as ⎞ ⎛ [Q, P]b exp ⎝i ψ Q (x1 , x2 , . . 71) j=1 where the “boson symbol” [Q, P]b means [Q, P]b = sign(P)A(k P1 , k P2 , . . , k P N ). 63). The formal structure of the Bethe ansatz remains unchanged for many other integrable Bose systems; only the two-body phase shift θ(k) depends on the particular form of the particle interaction.
E. I1 < I2 . 51) +A(312)ei(k3 x1 +k1 x2 +k2 x3 ) − A(321)ei(k3 x1 +k2 x2 +k1 x3 ) . The corresponding energy is given by E = k12 +k22 +k32 . 44). 41) taken at x2 = x3 implies that A(132) = e−iθ23 , A(123) A(231) = e−iθ13 , A(213) A(321) = e−iθ12 . 54) etc. The formal structure of the A-coefficients is obvious. Another possible representation of the coefficients is A(P) = (k P1 − k P2 + ic)(k P1 − k P3 + ic)(k P2 − k P3 + ic). 55) k1 17), we can write J k = J (I + G)ε /2 = −Gε /2 = ε /2 − k. Consequently, n = 2qρ(q) + ρ(q) ε (−q) ε (q) −q − +q 2 2 = ρ(q)ε (q). 4) The second derivative of the pressure with respect to the chemical potential can be calculated as follows ∂ ∂n ∂2 P = = 2 ∂μ ∂μ ∂μ q −q dk ρ(k) = 2ρ(q) ∂q + ∂μ q dk −q ∂ρ(k) . 5) To obtain an equation for ∂ρ(k)/∂μ, we differentiate Eq. 2) with respect to μ, with the result (I + G) ∂q ∂ρ (k) = −ρ(q) [G(k − q) + G(k + q)]. 6) Application of the operator I + J to both sides of this equation leads to ∂q ∂ρ(k) = ρ(q) [J (k, q) + J (k, −q)].