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 $D_d$ acting on the Hilbert space $\big(\ell^2(\mathbb{Z};\mathbb{C})\big)^2$, 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 $\ell^2(\mathbb{Z};\mathbb{C})$. 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 $[D_d, D_d^†] = -2mc^3(-Δ_{\mathrm{disc}})σ_2$, 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 $\big(\ell^2(h\mathbb{Z};\mathbb{C})\big)^2$ isometrically into $\big(L^2(\mathbb{R};\mathbb{C})\big)^2$ 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 $\mathcal{O}(h)$ as $h \to 0^+$, without requiring frequency cutoffs.