Spectral Theory and Continuum Norm Resolvent Convergence of a One-Sided Discrete Dirac-Type Operator
Abstract
We investigate a one-dimensional discrete Dirac-type operator Dd acting on the Hilbert space (ℓ2(ℤ;ℂ))2, obtained from the classical continuous free Dirac operator by replacing the spatial derivative with the one-sided backward difference operator I − S1, where S1 denotes the right shift on ℓ2(ℤ;ℂ). We establish the fundamental operator-theoretic and spectral properties of Dd: unlike symmetric or supersymmetric lattice discretizations, Dd is bounded and non-normal for every non-zero rest mass m > 0, with self-commutator [Dd, Dd†] = −2mc3(−Δdisc)σ2, yet it is quadratically normal. We compute the exact operator norm, determine the exact spectral radius in two mass regimes, and show via Fourier--Laurent analysis that the spectrum σ(Dd) is purely continuous and lies on a two-component algebraic curve in the complex plane undergoing a topological transition at mc2 = 2c. Finally, introducing a lattice spacing h > 0 and embedding (ℓ2(hℤ;ℂ))2 isometrically into (L2(ℝ;ℂ))2 via Whittaker--Shannon band-limited interpolation, we prove that the scaled non-normal discrete Dirac-type operator Dd,h converges to the continuous self-adjoint Dirac operator D in the norm resolvent sense at the optimal rate 𝒪(h) as h → 0+, without requiring frequency cutoffs.
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BibTeX
@misc{niwinski2026spectral,
title = {{Spectral Theory and Continuum Norm Resolvent Convergence of a One-Sided Discrete Dirac-Type Operator}},
author = {Mariusz Niwiński},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.00820},
eprint = {2611.00820},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.00820},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Spectral Theory and Continuum Norm Resolvent Convergence of a One-Sided Discrete Dirac-Type Operator
AU - Mariusz Niwiński
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.00820
UR - https://arxiv.org/abs/2611.00820
AB - We investigate a one-dimensional discrete Dirac-type operator D_d acting on the Hilbert space (ℓ^2(ℤ;ℂ))^2, obtained from the classical continuous free Dirac operator by replacing the spatial derivative with the one-sided backward difference operator I − S_1, where S_1 denotes the right shift on ℓ^2(ℤ;ℂ). We establish the fundamental operator-theoretic and spectral properties of D_d: unlike symmetric or supersymmetric lattice discretizations, D_d is bounded and non-normal for every non-zero rest mass m > 0, with self-commutator [D_d, D_d^†] = −2mc^3(−Δ_disc)σ_2, yet it is quadratically normal. We compute the exact operator norm, determine the exact spectral radius in two mass regimes, and show via Fourier--Laurent analysis that the spectrum σ(D_d) is purely continuous and lies on a two-component algebraic curve in the complex plane undergoing a topological transition at mc^2 = 2c. Finally, introducing a lattice spacing h > 0 and embedding (ℓ^2(hℤ;ℂ))^2 isometrically into (L^2(ℝ;ℂ))^2 via Whittaker--Shannon band-limited interpolation, we prove that the scaled non-normal discrete Dirac-type operator D_d,h converges to the continuous self-adjoint Dirac operator D in the norm resolvent sense at the optimal rate 𝒪(h) as h → 0^+, without requiring frequency cutoffs.
ER -