A note on adiabatic invariance in Hamiltonian systems depending singularly on the slow time

A one-degree-of-freedom oscillatory Hamiltonian system with a parameter depending singularly on the slow time is considered. It is shown that the system possesses an adiabatic invariant and its asymptotics is estimated for a rather general type of singularity. The leading term of the asymptotics turns out to be given by an integral of Fresnel type and the order of asymptotics is related to the type of singularity (stronger singularities cause larger change of adiabatic invariant). The result is applied to estimate the reflection coefficient in the problem of scattering electromagnetic wave on an obstacle with refraction index depending singularly on the coordinate. The relation to the stationary phase method is outlined.