Computing Spectral Measures and Spectral Types: New Algorithms and Classifications

08/19/2019 ∙ by Matthew John Colbrook, et al. ∙ 0

Despite new results on computing the spectrum, there has been no general method able to compute spectral measures (as given by the classical spectral theorem) of infinite-dimensional normal operators. Given a matrix representation, we show that if each matrix column decays at infinity at a known asymptotic rate, then it is possible to compute spectral measures of self-adjoint and unitary linear operators on separable Hilbert spaces. The central ingredient of the new algorithm is the computation of the resolvent operator with error control. Computational spectral problems in infinite dimensions have led to the SCI hierarchy, which classifies the difficulty of a problem through the number of limits needed to numerically compute the solution. We classify the computation of measures, measure decompositions, types of spectra (pure point, absolutely continuous, singular continuous), functional calculus and Radon--Nikodym derivatives in the SCI hierarchy for such operators. The new algorithms are demonstrated to be efficient and practical on examples taken from orthogonal polynomials on the real line and the unit circle, and are also applied to evolution equations on a two-dimensional quasicrystal model.

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