Exact energy distribution function in time-dependent harmonic oscillator

Following a recent work by Robnik and Romanovski (J.Phys.A: Math.Gen. 39 (2006) L35, Open Syst. & Infor. Dyn. 13 (2006) 197-222) we derive the explicit formula for the universal distribution function of the final energies in a time-dependent 1D harmonic oscillator, whose functional form does not depend on the details of the frequency ω(t), and is closely related to the conservation of the adiabatic invariant. The normalized distribution function is P (x) = π(2μ − x) 12 , where x = E1 − Ē1, E1 is the final energy, Ē1 is its average value, and μ is the variance of E1. Ē1 and μ 2 can be calculated exactly using the WKB approach to all orders. PACS numbers: 05.45.-a, 45.20.-d, 45.30.+s, 47.52.+j In a recent work [1, 2] Robnik and Romanovski have studied the energy evolution in a general 1D time-dependent harmonic oscillator and studied the closely related questions of the conservation of adiabatic invariants [3, 4, 5, 6, 7]. Starting with the ensemble of uniformly distributed (w.r.t. the canonical angle variable) initial conditions on the initial invariant torus of energy E0, they have calculated the average final energy Ē1, the variance μ 2 and all higher moments. The even moments are powers of μ, whilst the odd moments are exactly zero, because the distribution function P (E1) of the final energies E1 is an even function w.r.t. Ē1. In this Letter we derive explicit formula for P (E1), namely we shall derive P (E1) = Re 1 π √ 2μ2 − x2 , (1) where x = E1 − Ē1, and Re denotes the real part, so that (1) is zero for |x| > μ √ 2. This we do by using the exact results for the higher (even) moments and by employing the characteristic function f(y) of P (x). Letter to the Editor 2 The dynamics of our system is described by the Newton equation q̈ + ω(t)q = 0 (2) which is generated by the system’s Hamilton function H = H(q, p, t), whose numerical value E(t) at time t is precisely the total energy of the system at time t, and in case of 1D harmonic oscillator this is H = p 2M + 1 2 Mω(t)q, (3) where q, p,M, ω are the coordinate, the momentum, the mass and the frequency of the linear oscillator, respectively. The dynamics is linear in q, p, as described by (2), but nonlinear as a function of ω(t) and therefore is subject to the nonlinear dynamical analysis. By using the index 0 and 1 we denote the initial (t = t0) and final (t = t1) value of the variables. The transition map Φ maps initial conditions (q0, p0) onto the final conditions (q1, p1) Φ : (