Convergence Rates and Explicit Error Bounds of Hill's Method for Spectra of Self-Adjoint Differential Operators

TL;DR

This paper analyzes the convergence rates and provides explicit error bounds for Hill's method when computing spectra of self-adjoint differential operators with periodic coefficients, supported by numerical examples.

Contribution

It establishes convergence rates and explicit error bounds for Hill's method specifically for self-adjoint operators, with verifiable conditions using Gershgorin's theorem.

Findings

Convergence rates depend on the dimension of the approximation.

Explicit error bounds are derived under certain conditions.

Numerical examples confirm theoretical results.

Abstract

We present the convergence rates and the explicit error bounds of Hill's method, which is a numerical method for computing the spectra of ordinary differential operators with periodic coefficients. This method approximates the operator by a finite dimensional matrix. On the assumption that the operator is selfadjoint, it is shown that, under some conditions, we can obtain the convergence rates of eigenvalues with respect to the dimension and the explicit error bounds. Numerical examples demonstrate that we can verify these conditions using Gershgorin's theorem for some real problems. Main theorems are proved using the Dunford integrals which project an eigenvector to the corresponding eigenspace.