ADIABATIC INVARIANT OF THE HARMONIC-OSCILLATOR, COMPLEX MATCHING AND RESURGENCE

The linear oscillator equation with a frequency slowly dependent on ti me is used to test a method to compute exponentially small quantities. This work presents the matching method in the complex plane as a tool to obtain rigorously the asymptotic variation of the action of the as sociated Hamiltonian beyond all orders. The solution in the complex pl ane is approximated by a series in which all terms present a singulari ty at the same point. Following matching techniques near this singular ity one is led to an equation which does not depend on any parameter, the so-called inner equation, of a Riccati-type. This equation is stud ied by resurgence methods.