EXACT ANALYSIS OF ADIABATIC INVARIANTS IN TIME DEPENDENT HARMONIC OSCILLATOR

The theory of adiabatic invariants has a long history, and very important implications and applications in many different branches of physics, classically and quantally, but is rarely founded on rigorous results. Here we treat the general time-dependent one-dimensional linear (harmonic) oscillator, whose Newton equation <(q)double over dot> + omega(2)(t)q = 0 cannot be solved in general. We follow the time-evolution of an initial ensemble of phase points with sharply defined energy E(0) at time t = 0 (microcanonical ensemble) and calculate rigorously the distribution of energy E(1) after time t = T, which is fully (all moments, including the variance mu(2)) determined by the first moment (E) over bar (1). For example, mu(2) = E(0)(2[()(E) over bar (1)/E(0))(2) - (omega(T)/omega(0))(2)]/2, and all higher even moments are powers of mu(2), whilst the odd ones vanish identically. This distribution function does not depend on any further details of the function omega(t) and is in this sense universal, it is a normalized distribution function given by P(x) = pi(-1)(2 mu(2) - x(2))(-1/2), where x = E(1) - (E) over bar (1). (E) over bar (1) and mu(2) can be calculated exactly in some cases. In ideal adiabacity (E) over bar (1) = omega(T)E(0)/omega(0), and the variance mu(2) is zero, whilst for infinite T we calculate (E) over bar (1), and mu(2) for the general case using exact WKB-theory to all orders. We prove that if omega(t) is of class C(m) (all derivatives up to and including the order m are continuous) mu(2) proportional to T(-2(m+1)), whilst for class C(infinity) it is known to be exponential mu(2) proportional to exp(-alpha T). Due to the positivity of mu(2) we also see that the adiabatic invariant l = (E) over bar (1)/omega(T) at the average energy (E) over bar (1) never decreases,