Continuum limits for discrete Dirac operators on 2D square lattices

Schmidt, Karl Michael ORCID: https://orcid.org/0000-0002-0227-3024 and Umeda, Tomio 2023. Continuum limits for discrete Dirac operators on 2D square lattices. Analysis and Mathematical Physics 13 (46) 10.1007/s13324-023-00809-7

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Abstract

We discuss the continuum limit of discrete Dirac operators on the square lattice in R2 as the mesh size tends to zero. To this end, we propose the most natural and simplest embedding of ℓ2(Zdh) into L2(Rd) , which enables us to compare the discrete Dirac operators with the continuum Dirac operators in the same Hilbert space L2(R2)2 . In particular, we prove that the discrete Dirac operators converge to the continuum Dirac operators in the strong resolvent sense. Potentials are assumed to be bounded and uniformly continuous functions on R2 and allowed to be complex matrix-valued. We also prove that the discrete Dirac operators do not converge to the continuum Dirac operators in the norm resolvent sense. This is closely related to the observation that the Liouville theorem does not hold in discrete complex analysis.

Item Type:	Article
Date Type:	Published Online
Status:	Published
Schools:	Mathematics
Publisher:	Springer Verlag (Germany)
ISSN:	1664-2368
Date of First Compliant Deposit:	12 May 2023
Date of Acceptance:	26 April 2023
Last Modified:	18 May 2023 09:17
URI:	https://orca.cardiff.ac.uk/id/eprint/159201

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