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Convergence of Eigenvector Continuation.

Avik Sarkar1, Dean Lee1

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Summary
This summary is machine-generated.

Eigenvector continuation, a method for finding eigenvalues and eigenvectors, has mysterious convergence properties. This study analyzes its convergence, revealing it generally surpasses perturbation theory by mitigating differential folding.

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Area of Science:

  • Computational physics
  • Quantum mechanics
  • Numerical analysis

Background:

  • Eigenvector continuation is an efficient computational method for determining extremal eigenvalues and eigenvectors of Hamiltonian matrices.
  • The method's rapid convergence properties and underlying mechanisms remain largely unstudied and mysterious.
  • Existing knowledge lacks a deep understanding of how eigenvector continuation converges.

Purpose of the Study:

  • To investigate and elucidate the convergence properties of eigenvector continuation.
  • To provide the first mathematical analysis of eigenvector continuation's convergence behavior.
  • To compare the convergence of eigenvector continuation with traditional perturbation theory.

Main Methods:

  • Introduction of a novel variant, 'vector continuation', for mathematical analysis.
  • Proof of identical convergence properties between eigenvector continuation and vector continuation.
  • Analysis of vector continuation's convergence to understand the parent method.

Main Results:

  • Eigenvector continuation generally exhibits faster convergence than perturbation theory.
  • The enhanced convergence is attributed to the elimination of 'differential folding', an interference phenomenon in perturbation theory.
  • The study predicts convergence behavior both within and beyond the radius of convergence of perturbation theory.

Conclusions:

  • The convergence rate of eigenvector continuation can be determined from power series expansions, despite being a nonperturbative method.
  • The analysis provides new insights into the nature of divergences encountered in perturbation theory.
  • This work demystifies the convergence of eigenvector continuation, offering a predictive framework.