Approach to energy eigenvalues and eigenfunctions from nonperturbative regions of eigenfunctions

Area of Science:

• Quantum Mechanics
• Condensed Matter Physics
• Computational Physics

Background:

• Hamiltonian matrices describe quantum systems.
• Diagonalization of truncated matrices is a common approximation method.
• Perturbation theory is used to refine approximations.

Purpose of the Study:

• To determine the optimal size of truncated matrices for accurate energy eigenvalue and eigenfunction approximation.
• To establish a method for estimating the required matrix size before full eigenfunction calculation.

Main Methods:

• Generalization of Brillouin-Wigner perturbation theory.
• Numerical validation using the Wigner-band random-matrix model.
• Analysis of the relationship between truncated matrix size and nonperturbative regions.

Main Results:

• Truncated matrix size must exceed nonperturbative regions by several Hamiltonian matrix band widths for accurate approximations.
• Nonperturbative regions can be estimated prior to full eigenfunction computation.
• Numerical results confirm that ~99% of eigenfunctions are obtained when matrix size exceeds nonperturbative regions by three band widths.

Conclusions:

• The study provides a guideline for selecting appropriate truncated matrix sizes in quantum mechanical calculations.
• The findings improve the efficiency and accuracy of approximating energy eigenvalues and eigenfunctions.
• The generalized perturbation theory offers a robust framework for analyzing Hamiltonian matrices.