Norm resolvent convergence of discretized Fourier multipliers

Horia Cornean, Henrik Garde*, Arne Jensen

*Corresponding author for this work

Research output: Contribution to journal/Conference contribution in journal/Contribution to newspaperJournal articleResearchpeer-review

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 pseudo-relativistic free Hamiltonian. The potentials are real, bounded, and Hölder continuous. As a side-product, 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.

Original languageEnglish
Article number71
JournalJournal of Fourier Analysis and Applications
Volume27
Issue4
Number of pages31
ISSN1069-5869
DOIs
Publication statusPublished - Aug 2021

Keywords

  • Norm resolvent convergence
  • Fourier multiplier
  • Lattice
  • Hausdorff distance

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