On the quality of complementary bounds for eigenvalues

TL;DR

This paper analyzes the Lehmann-Maehly-Goerisch method for estimating eigenvalues of semi-definite self-adjoint operators, providing convergence rates, optimal shift parameter choices, and numerical validation.

Contribution

It offers a concrete formulation of the method, derives precise convergence rates based on trial space quality, and investigates the optimal shift parameter for improved bounds.

Findings

Convergence rates depend on how well trial spaces approximate spectral subspaces.

Optimal shift parameter enhances the quality of eigenvalue bounds.

Numerical experiments validate theoretical convergence and optimality results.

Abstract

A concrete formulation of the Lehmann-Maehly-Goerisch method for semi-definite self-adjoint operators with compact resolvent is considered. Precise rates of convergence are determined in terms of how well the trial spaces capture the spectral subspace of the operator. Optimality of the choice of a shift parameter which is intrinsic to the method is also examined. The main theoretical findings are illustrated by means of a few numerical experiments involving one-dimensional Schrodinger operators.