Uniform Asymptotics in the Problem of Superfluidity of Classical Liquids in Nanotubes

In the preceding papers (see [1, 2]), the superfluidity of the classical liquid was proved under the assumption that the parameters N and r, where N is the particle number and r it the capillary radius, tend respectively to infinity and to zero so that 1 N ≪ r R , where R is the capillary length. In the present paper, this assumption is removed. 1. We first note that solutions of the variational equation for the Vlasov equation do not coincide with the classical limit for the variational equations corresponding to the mean-field equations in quantum theory. We consider mean-field equations of the form ih ∂ ∂t φ(x) = ( − h 2 2m ∆+Wt(x) ) φ(x), Wt(x) = U(x) + ∫ V (x, y)|φ(y)|dy, (1) with the initial condition φ|t=0 = φ0, where φ0 ∈W∞ 2 (R), ∫ dx|φ0(x)| = 1. To find an asymptotic representation of the complex-germ type [3], we consider a system consisting of the Hartree equation and its conjugate equation. We then take the system of variational equations for it and replace the variations δφ and δφ with the independent functions F and G. For F and G, we obtain the system of equations: i ∂F (x) ∂t = ∫ dy ( δH δφ∗(x)δφ(y) F (y) + δH δφ∗(x)δ∗φ(y) G(y) ) ; (2) −i (x) ∂t = ∫ dy ( δH δφ(x)δφ(y) F (y) + δH δφ(x)δ∗φ(y) G(y) ) . Roughly speaking, the classical equations can be obtained from the quantum ones using a substitution of the form φ = χe i h S (the WKB method), φ = χe i h S , S = S, χ = χ(x, t) ∈ C, S = S(x, t) ∈ C. For variational equations, it is natural to vary not only the limit equation for χ and χ but also the functions S and S. This gives an important new term in the solution of the equation for collective oscillations. We consider this fact for the simplest example investigated in the famous work by Bogoliubov on a weakly nonideal Bose gas [4]. Let U = 0 for (1) in a three-dimensional box with edge length L, and let the Lperiodicity condition be imposed on the wave functions in this case (i.e., a problem on the torus with generators L,L, and L is considered). Then the function