SpecSolve: Spectral methods for spectral measures

TL;DR

SpecSolve introduces advanced spectral methods to compute spectral measures of complex operators, extending its capabilities to singular integro-differential operators and operator pencils with high accuracy and flexibility.

Contribution

The paper extends the SpecSolve package to handle singular integro-differential operators and operator pencils using various spectral methods, enhancing its computational power.

Findings

Achieves high-order convergence in spectral measure computation.

Successfully extends to singular integro-differential operators.

Demonstrates flexibility with multiple spectral methods.

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-the-shelf 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.