On the convergence of the quadratic method

TL;DR

This paper analyzes the convergence properties of the quadratic method for eigenvalue enclosures, providing improved asymptotic bounds and illustrating the theory with numerical experiments on Schrödinger operators.

Contribution

It offers new, explicit asymptotic bounds for convergence of the quadratic method, improving upon previous results for self-adjoint operators.

Findings

Derived explicit asymptotic bounds for convergence.

Demonstrated improved bounds through numerical experiments.

Validated the theory with benchmark Schrödinger operator models.

Abstract

The convergence of the so-called quadratic method for computing eigenvalue enclosures of general self-adjoint operators is examined. Explicit asymptotic bounds for convergence to isolated eigenvalues are found. These bounds turn out to improve significantly upon those determined in previous investigations. The theory is illustrated by means of several numerical experiments performed on particularly simple benchmark models of one-dimensional Schrodinger operators.