Computing spectral measures of self-adjoint operators

TL;DR

This paper introduces a high-order convergent algorithm using the resolvent operator to compute spectral measures of self-adjoint operators with applications to differential, integral, and lattice operators, achieving high accuracy and avoiding spectral pollution.

Contribution

It develops a novel algorithm based on the resolvent operator for computing spectral measures with high-order convergence and provides explicit error bounds and practical implementations.

Findings

Achieves arbitrarily high-order convergence in spectral measure approximation.

Provides explicit pointwise and $L^p$ error bounds based on measure regularity.

Successfully computes eigenvalues with near machine precision without spectral pollution.

Abstract

Using the resolvent operator, we develop an algorithm for computing smoothed approximations of spectral measures associated with self-adjoint operators. The algorithm can achieve arbitrarily high-orders of convergence in terms of a smoothing parameter for computing spectral measures of general differential, integral, and lattice operators. Explicit pointwise and $L^p$-error bounds are derived in terms of the local regularity of the measure. We provide numerical examples, including a partial differential operator, a magnetic tight-binding model of graphene, and compute one thousand eigenvalues of a Dirac operator to near machine precision without spectral pollution. The algorithm is publicly available in $\texttt{SpecSolve}$, which is a software package written in MATLAB.