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Self-consistent theory of lower bounds for eigenvalues.

Eli Pollak1, Rocco Martinazzo2

  • 1Chemical and Biological Physics Department, Weizmann Institute of Science, 76100 Rehovot, Israel.

The Journal of Chemical Physics
|July 3, 2020
PubMed
Summary
This summary is machine-generated.

A new theory provides practical methods for calculating lower bounds to eigenvalues of Hermitian operators. This approach, using the Lanczos method and self-consistent refinement, offers accurate eigenvalue computations.

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Area of Science:

  • Quantum mechanics
  • Computational physics
  • Numerical analysis

Background:

  • Calculating eigenvalues for Hermitian operators is crucial in quantum mechanics.
  • Existing methods often provide upper bounds, but lower bounds are also essential for rigorous analysis.

Purpose of the Study:

  • To present a rigorous and practically applicable theory for obtaining lower bounds to eigenvalues of Hermitian operators.
  • To develop algorithms for computing residual energies essential for eigenvalue calculations.

Main Methods:

  • Development of a self-consistent theory for eigenvalue lower bound computation.
  • Utilizing the Lanczos method to create a tridiagonal representation of the operator.
  • Iterative refinement of lower bounds for improved accuracy.

Main Results:

  • Demonstrated a self-consistent theory where lower bounds for one state can improve bounds for others.
  • Achieved high accuracy in eigenvalue lower bound calculations, exemplified by a quartic oscillator model.
  • Reduced the relative error in the ground state lower bound to 6 × 10-6 with only five basis states.

Conclusions:

  • The presented theory offers a practical and rigorous method for computing lower bounds to eigenvalues.
  • The self-consistent nature of the method allows for iterative improvement of accuracy.
  • Lower bound computations are proposed as a potential staple in future eigenvalue calculations.