Julia sets in iterative KAM methods for eigenvalue problems
The authors present two versions of the KAM algorithm for quantum mechanics, which is based on the two iterative KAM methods for eigenvalue problems. The authors discuss the convergence of these iterations for 2, 3 and \(n\)-dimensional symmetric matrices with arbitrarily large perturbations. Note that the proofs of convergence of KAM methods are based on proving the contractivity of the map with respect to some norm. The presence of Julia sets and non monotonic behaviour are obstacles for proving the convergence in all regions where the iteration converges. The Julia set, on which the iteration does not converge depends on the choice of unitary transformation. The authors observe different domains separated by Julia sets related to avoided crossings.
scientific article; zbMATH DE number 1371613