Norm resolvent convergence of discretized Fourier multipliers
Abstract
We prove norm estimates for the difference of resolvents of operators and their discrete counterparts, embedded into the continuum using biorthogonal Riesz sequences. The estimates are given in the operator norm for operators on square integrable functions, and depend explicitly on the mesh size for the discrete operators. The operators are a sum of a Fourier multiplier and a multiplicative potential. The Fourier multipliers include the fractional Laplacian and the pseudorelativistic free Hamiltonian. The potentials are real, bounded, and Hölder continuous. As a sideproduct, the Hausdorff distance between the spectra of the resolvents of the continuous and discrete operators decays with the same rate in the mesh size as for the norm resolvent estimates. The same result holds for the spectra of the original operators in a local Hausdorff distance.
 Publication:

arXiv eprints
 Pub Date:
 October 2020
 DOI:
 10.48550/arXiv.2010.16215
 arXiv:
 arXiv:2010.16215
 Bibcode:
 2020arXiv201016215C
 Keywords:

 Mathematics  Functional Analysis;
 Mathematical Physics;
 Mathematics  Spectral Theory;
 47A10;
 47A58;
 42B15;
 (47A11;
 47B39)
 EPrint:
 26 pages