Computing spectral measures of self-adjoint operators. (English) Zbl 07379587
Summary: 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 and a magnetic tight-binding model of graphene, and compute 1000 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.
MSC:
| 47A10 | Spectrum, resolvent |
| 46N40 | Applications of functional analysis in numerical analysis |
| 47N50 | Applications of operator theory in the physical sciences |
| 65N35 | Spectral, collocation and related methods for boundary value problems involving PDEs |
| 81Q10 | Selfadjoint operator theory in quantum theory, including spectral analysis |
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