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Iterative eigenvalue method using the Bloch wave operator formalism with Padé approximants and absorbing boundaries.

Georges Jolicard1, David Viennot, John P Killingbeck

  • 1Laboratoire d'Astrophysique de l'Observatoire de Besançon (CNRS UMR 6091), 41 bis Avenue de l'Observatoire, Boîte Postale 1615, 25010 Besançon Cedex, France.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|December 17, 2004
PubMed
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This study introduces an iterative method for eigenvalue and eigenvector calculations in large non-Hermitian matrices. The technique enhances convergence for Floquet eigenstates using absorbing boundaries.

Area of Science:

  • Quantum mechanics
  • Computational physics
  • Matrix theory

Background:

  • Calculating eigenvalues and eigenvectors for large non-Hermitian matrices is computationally challenging.
  • Non-Hermitian matrices are relevant in various fields, including quantum mechanics and open systems.

Purpose of the Study:

  • To present a novel iterative method for efficiently computing eigenvalues and eigenvectors of large non-Hermitian matrices.
  • To improve the convergence properties of iterative calculations, particularly for systems with strongly coupled states and Floquet eigenstates.

Main Methods:

  • An iterative procedure is employed to solve the Bloch equation (HOmega=OmegaHOmega) from wave operator theory.
  • Nonlinear transformations, including diagonal element translation and Padé approximants, are used to handle intermediate strongly coupled states.

Related Experiment Videos

  • Time-dependent absorbing boundaries are incorporated for calculations involving Floquet eigenstates.

    • Main Results:

    • The proposed iterative method demonstrates effectiveness in calculating eigenvalues and eigenvectors for large non-Hermitian matrices.
    • The inclusion of time-dependent absorbing boundaries significantly enhances the convergence of iterative calculations for Floquet eigenstates.
    • The method successfully treats strongly coupled states within an intermediate space.

    Conclusions:

    • The presented iterative method offers an efficient approach for eigenvalue/eigenvector problems involving large non-Hermitian matrices.
    • The technique, especially with absorbing boundaries, provides a robust solution for complex systems like those described by Floquet theory.
    • This work contributes to advancing computational methods in quantum mechanics and related fields.