Computing spectral measures of self-adjoint operators

2020/06/02 by Colbrook, Matthew J., Horning, Andrew, Townsend, Alex · 2 citations
#46N40 #47A10 #47N50 #65N35 #81Q10 #FOS: Mathematics #FOS: Physical sciences #Functional Analysis (math.FA) #Mathematical Physics (math-ph) #Numerical Analysis (math.NA) #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.2006.01766

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 SpecSolve, which is a software package written in MATLAB.

Cited by

• A high-order regularized delta-Chebyshev method for computing spectral densities
• An Introductory Guide to Koopman Learning

Related

• SpecSolve: Spectral methods for spectral measures
• A new approach to spectral approximation
• Computing Spectral Measures and Spectral Types
• The Galerkin Method for Perturbed Self-Adjoint Operators and\n Applications
• Spectral Pollution