Julia sets in iterative KAM methods for eigenvalue problems
type: Item
instance of: scholarly article
title: Julia sets in iterative KAM methods for eigenvalue problems
zbMATH Open document ID: 0933.37052
DOI: 10.1016/S0960-0779(97)00187-2
author: S. H. Smith
published in: Chaos, Solitons and Fractals
publication date: 2000-04-09T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 37F50
Mathematics Subject Classification ID: 37J40
zbMATH DE Number: 1371613
zbMATH Keywords: convergence
zbMATH Keywords: KAM algorithm
zbMATH Keywords: eigenvalue problems
zbMATH Keywords: Julia sets
Wikidata QID: Q121036975
reviewed by: Messoud Efendiev
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1371613