Abstract

Unlike its matrix counterpart, the spectral measure of a self-adjoint operator may have an absolutely continuous component and an associated density function, e.g., in applications posed on unbounded domains. The state-of-the-art computational methods for these problems typically approximate the density function using a smoothed sum of Dirac measures, corresponding to the spectral measure of a matrix discretization of the operator. However, it is often difficult to determine the smoothing and discretization parameters that are necessary to accurately and efficiently resolve the density function. In this talk, we present an adaptive framework for computing the spectral measure of a self-adjoint operator that provides insight into the selection of smoothing and discretization parameters. We show how to construct local approximations to the density that converge rapidly when the density function is smooth and discuss possible connections with Pade approximation that could alleviate deteriorating convergence rates near non-smooth points.