Continuum limits for discrete Dirac operators on 2D square lattices

Schmidt, Karl Michael; Umeda, T.

doi:10.48550/arxiv.2109.04052

Cited by 1 publication

(3 citation statements)

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“…These latter results obviously imply strong convergence. In particular, we recover the result in [5] for d = 2 with the discretization H fb 0,h .…”

Section: Sobolev Space Estimates and Strong Convergencesupporting

confidence: 73%

“…This result implies that the strong convergence result in [5] cannot be improved to a norm convergence result, without modifying the discretization.…”

Section: The 2d Forward-backward Difference Modelmentioning

confidence: 99%

“…Note that the results in [1] do not directly apply to Dirac operators, since the free Dirac operators do not satisfy an essential symmetry condition [1, Assumption 3.1(4)]. In [5] Schmidt and Umeda proved strong resolvent convergence for Dirac operators in dimension d = 2 using the discretization H fb 0,h . They allow a class of bounded non-selfadjoint potentials and also state corresponding results for dimensions d = 1, 3.…”

Section: Introductionmentioning

confidence: 99%

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Discrete approximations to Dirac operators and norm resolvent convergence

Cornean¹,

Garde²,

Jensen³

2022

Preprint

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We consider continuous Dirac operators defined on R d , d ∈ {1, 2, 3}, together with various discrete versions of them. Both forward-backward and symmetric finite differences are used as approximations to partial derivatives. We also allow a bounded, Hölder continuous, and self-adjoint matrix-valued potential, which in the discrete setting is evaluated on the mesh. Our main goal is to investigate whether the proposed discrete models converge in norm resolvent sense to their continuous counterparts, as the mesh size tends to zero and up to a natural embedding of the discrete space into the continuous one. In dimension one we show that forward-backward differences lead to norm resolvent convergence, while in dimension two and three they do not. The same negative result holds in all dimensions when symmetric differences are used. On the other hand, strong resolvent convergence holds in all these cases. Nevertheless, and quite remarkably, a rather simple but non-standard modification to the discrete models, involving the mass term, ensures norm resolvent convergence in general.

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“…These latter results obviously imply strong convergence. In particular, we recover the result in [5] for d = 2 with the discretization H fb 0,h .…”

Section: Sobolev Space Estimates and Strong Convergencesupporting

confidence: 73%

“…This result implies that the strong convergence result in [5] cannot be improved to a norm convergence result, without modifying the discretization.…”

Section: The 2d Forward-backward Difference Modelmentioning

confidence: 99%