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