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RESEARCH PRODUCT

Domains of Convergence of Kam Type Iterations for Eigenvalue Problems

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The KAM technique was first introduced to deal with small denominator problems appearing in perturbation of invariant tori in classical mechanics [1, 2]. Similar methods were later applied to many different problems, like e.g. eigenvalue problems for time dependent problems in the Floquet representation [3, 4, 5, 6]. Most of the known results are valid for sufficiently small perturbation of some simple (integrable) system. The phenomena arising for large perturbations, in particular critical perturbations at which a given torus loses its stability, have been discussed in the framework of some approximate schemes inspired in renormalization group ideas [7, 8, 9]. In this framework, an iteration is constructed that, involves a KAM transformation combined with some mapping of resonances and some rescaling. The general idea is that for small perturbations the iteration should converge to a trivial fixed point (which is associated with the existence of a smooth invariant torus); if a critical perturbation is attained, at which the torus starts breaking up, the iteration is expected to converge to a different “non trivial” fixed point. A precise mathematical implementation of this picture has however not yet been attained.