SpecSolve: Spectral methods for spectral measures

Abstract

Self-adjoint operators on infinite-dimensional spaces with continuous spectra are abundant but do not possess a basis of eigenfunctions. Rather, diagonalization is achieved through spectral measures. The SpecSolve package [SIAM Rev., 63(3) (2021), pp. 489-524] computes spectral measures of general (self-adjoint) differential and integral operators by combining state-of-the-art adaptive spectral methods with an efficient resolvent-based strategy. The algorithm achieves arbitrarily high orders of convergence in terms of a smoothing parameter, allowing computation of both discrete and continuous spectral components. This article extends SpecSolve to two important classes of operators: singular integro-differential operators and general operator pencils. Essential computational steps are performed with off-theshelf spectral methods, including spectral methods on the real line, the ultraspherical spectral method, Chebyshev and Fourier spectral methods, and the (ℎ𝑝-adaptive and sparse) ultraspherical spectral element method. This collection illustrates the power and flexibility of SpecSolve's "discretization-oblivious" paradigm.