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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bambusi, D., Penati, T.: Continuous approximation of breathers in one- and two-dimensional DNLS lattices. Nonlinearity 23, 143–157 (2010)
Christensen, O.: An Introduction to Frames and Riesz Bases, 2nd edn. Birkhäuser, Basel (2016)
Helffer, B., Sjöstrand, J.: Équation de Schrödinger avec champ magnétique et équation de Harper. In: H. Holden and A. Jensen (eds) Schrödinger Operators (Sønderborg, 1989). Lecture Notes in Phys., vol. 345, pp. 118–197. Springer-Verlag, Berlin (1988)
Hong, Y., Yang, C.: Strong convergence for discrete nonlinear Schrödinger equations in the continuum limit. SIAM J. Math. Anal. 51(2), 1297–1320 (2019)
Isozaki, H., Jensen, A.: Continuum limit for lattice Schrödinger operators. To appear in Rev. Math. Phys
Kato, T.: Perturbation Theory for Linear Operators, 2nd edn. Springer-Verlag, Berlin-New York (1976)
Meyer, Y.: Ondelettes et Opérateurs. I. Ondelettes. Hermann, Paris (1990)
Nakamura, S., Tadano, Y.: On a continuum limit of discrete Schrödinger operators on square lattice. J. Spectr. Theory 11(1), 355–367 (2021)
Rabinovich, V.: Wiener algebra of operators on the lattice \((\mu \mathbb{Z})^n\) depending on a small parameter \(\mu >0\). Complex Var. Elliptic Equ. 58(6), 751–766 (2013)
Reed, M., Simon, B.: Methods of Modern Mathematical Physics. I. Functional Analysis, Second edn. Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York (1980)
Acknowledgements
This research is partially supported by grant 8021–00084B from Independent Research Fund Denmark | Natural Sciences.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by Gabriela Steidl.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Cornean, H., Garde, H. & Jensen, A. Norm Resolvent Convergence of Discretized Fourier Multipliers. J Fourier Anal Appl 27, 71 (2021). https://doi.org/10.1007/s00041-021-09876-5
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00041-021-09876-5