Analysis and numerical methods in solid state physics and chemistry
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In the first part of the paper, we consider an atomic model of deposition over a quasi-periodic medium, that is, a quasi-periodic version of the well-known Frenkel-Kontorova model. We consider the problem of whether there are quasi-periodic equilibria with a frequency that resonates with the frequencies of the medium. We show that there are always perturbative expansions. We also prove a KAM theorem in a-posterori form. We show that if there is an approximate solution of the equilibrium equation satisfying non-degeneracy conditions, we can adjust one parameter and obtain a true solution which is close to the approximate solution. The proof is based on an iterative method of the KAM type and a novel technique of supplementing the equilibrium equation with another equation that factors the linearization of equilibrium equation. In the second part of the paper, we consider a model related to Transition State Theory in chemical reactions. We consider a particle with an initial position on the reactant side of a time-dependent energy barrier and study the invariant manifolds and associated bundles using the parameterization method.