Weak convergence methods and adiabatic results in classical and quantum mechanics

We will show that weak convergence methods are conveniently suited to explicitly access singular limits of a certain family of mechanical systems with multiple time scales. This family turns out to be characterized by the existence of sufficiently many adiabatic invariants.
The key step is the idenfication of the weak limits of all those quadratic quantities which carry important information for the limit system. This idenfication becomes possible by a weak convergence analogue of the Virial Theorem, resulting in certain matrix commutation relations.
We will address natural mechanical systems with strong constraining potentials and the adiabatic theorem of quantum mechanics. These examples will show that our approach considerably extends the possibility of passing through resonances.