Spectral stability and semiclassical measures for renormalized KAM systems

A semiclassical version of the classical KAM theorem about perturbations of constant vector fields on the torus is obtained. Moreover, given a small and bounded perturbation of a linear Hamiltonian on the torus with constant coefficients, the problem of finding an integrable counterterm that renormalizes the system making it canonically conjugate to the unperturbed one is also addressed in the semiclassical setting. These results are used to obtain a characterization of the sets of semiclassical measures and quantum limits associated to sequences of $L^2$-densities of eigenfunctions for the considered Schr\"odinger operators.