Spectral Measures of Self-Adjoint Operators

Chapter 4 shows how spectral measures generalise diagonalisation for self-adjoint operators with continuous spectrum and provides practical, provably convergent methods for computing them. Using the spectral theorem, we introduce projection-valued and scalar measures, their Lebesgue decomposition, and analyse how discretisations produce discrete measures. Strong and strong-resolvent convergence conditions ensure weak convergence of the measures and convergence of the functional calculus, clarifying spectral invisibility and pollution. From Stone’s formula, we derive resolvent-based algorithms: spectral projections are approximated by kernel convolutions (Poisson smoothing) evaluated by solving linear systems with adaptive error control as the smoothing parameter shrinks. This yields SCI-hierarchy classifications for setwise computation, functional calculus, and weak convergence. Finally, we accelerate Stone’s formula with high-order convolution and rational kernels, reducing the computational burden, prove convergence rates in several settings, and illustrate the approach on magneto-graphene, Dirac operators, and pseudodifferential operators related to internal waves.