  In this work, concentration properties of quasimodes for perturbed
semiclassical harmonic oscillators are studied. The starting point of this
research comes from the fact that, in the presence of resonances between
frequencies of the harmonic oscillator and for a generic bounded perturbation,
the set of semiclassical measures for quasimodes of sufficiently small width is
strictly smaller than the corresponding set of semiclassical measures for the
unperturbed system. In this work we precise the description of this set taking
into account the Diophantine properties of the vector of frequencies, the
separation between clusters of eigenvalues in the spectrum produced by the
presence of a perturbation, and the dynamical properties of the classical
Hamiltonian flow generated by the average of the symbol of the perturbation by
the harmonic oscillator flow. In particular, for the perturbed two-dimensional
periodic harmonic oscillator, we characterize the set of semiclassical measures
of quasimodes with a width that is critical for this problem. Two of the main
ingredients in the proof of these results, which are of independent interest,
are: (i) a novel construction of quasimodes based on propagation of coherent
states by several commuting flows and (ii) a general quantum Birkhoff normal
form for harmonic oscillators.
