Adiabatic Invariant of the Harmonic Oscillator, Complex Matching and Resurgence

The linear oscillator equation with a frequency slowly dependent on time 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 associated Hamiltonian beyond all orders. The solution in the complex plane is approximated by a series in which all terms present a singularity at the same point. Following matching techniques near this singularity 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 studied by resurgence methods.