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arXiv · 2006.01766

Computing spectral measures of self-adjoint operators

Matthew J. Colbrook·Andrew Horning·Alex Townsend
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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 $\texttt{SpecSolve}$, which is a software package written in MATLAB.

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Matthew J. Colbrook, Andrew Horning, Alex Townsend. 2020-06-02. Computing spectral measures of self-adjoint operators. https://arxiv.org/abs/2006.01766

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