On quasimodes of small diffusion operators corresponding to stable invariant tori with nonregular neighborhoods

Smooth vector fields V in $\[$\mathbb{R}$$ ⁿ having a k-dimensional invariant torus are considered and the possibility of constructing the asymptotic eigenfunctions localized in the neighborhood of this torus is analyzed for the case of a corresponding small diffusion operator V·∇−εΔ, ε>0. The asymptotic stability property of the invariant torus is a sufficient condition for the existence of such eigenfunctions. Both the regular case (where the variational system is reducible) and the nonregular case (where the variational system is nonreducible) are considered.