Nonlocal Generalized Dirac Oscillators in (1 + 1) Dimensions

We propose a nonlocal extension of the generalized Dirac oscillator (GDO) in $(1+1)$ dimensions by replacing the multiplicative interaction $f(x)$ with an integral operator $\hat F$ with kernel $f(x,x’)$. The resulting Dirac equation preserves an operator factorization and decouples into two nonlocal Schrödinger-type (Sturm–Liouville) equations for the spinor components. We derive explicit expressions for the associated supersymmetric partner kernels in terms of $f$ and its derivatives, and we show that a complex-translation metric $η=e^{-θp_x}$ leads to a simple sufficient \emph{kernel-level} pseudo-Hermiticity constraint, $f(x+\ii\hbarθ,x’+\ii\hbarθ)=f^*(x’,x)$, extending the familiar local complex-shift criteria. To provide a transparent \emph{nonlocal-to-local} interpretation, we adapt the Coz–Arnold–MacKellar current-based localization to each component equation, obtaining energy-dependent equivalent local potentials and multiplicative Perey (damping) factors. The mapping breaks down precisely at current zeros, thereby diagnosing the spurious solutions of the corresponding nonlocal Schrödinger problem. Finally, we illustrate the formalism with analytically tractable benchmarks (the local Dirac oscillator and a translation-invariant kernel) and with a finite-rank separable model (Gaussian form factor) that reduces the integro-differential problem to a small set of coupled ordinary differential equations and algebraic constraints.

💡 Research Summary

The paper introduces a non‑local extension of the generalized Dirac oscillator (GDO) in one spatial dimension, replacing the usual multiplicative “superpotential’’ f(x) by an integral operator ĤF with kernel f(x,x′). The authors show that, despite this non‑locality, the Dirac Hamiltonian retains a factorised form and the two spinor components decouple into a pair of non‑local Schrödinger‑type (Sturm–Liouville) equations. By defining non‑local ladder operators A = pₓ − iĤF and A† = pₓ + iĤF, they obtain second‑order equations A†A ψ₁ = ε ψ₁ and AA† ψ₂ = ε ψ₂, with ε = (E² − m²c⁴)/(c²). In coordinate space the corresponding kernels V₁, V₂ act as supersymmetric partners and are given explicitly by

 V₁,₂(x,x′) = (f ⋆ f)(x,x′) ∓ ħ(∂ₓ + ∂ₓ′) f(x,x′),

where (f ⋆ f)(x,x′) = ∫dy f(x,y) f(y,x′). In the local limit f(x,x′)=f(x)δ(x−x′) this reduces to the familiar V±(x)=f²(x)±ħ f′(x).

A central result is the kernel‑level pseudo‑Hermiticity condition for the complex‑translation metric η = e^{‑θpₓ}. Acting on wavefunctions, η shifts the arguments by iħθ, so that ηĤFη⁻¹ has kernel f(x+iħθ, x′+iħθ). The Hamiltonian satisfies H† = ηHη⁻¹ provided

 f(x+iħθ, x′+iħθ) = f* (x′, x).

This condition interpolates between ordinary Hermiticity (θ = 0) and the local complex‑shift condition f(x+iħθ)=f*(x) used in PT‑symmetric Dirac oscillators. Thus the authors obtain a simple sufficient criterion for pseudo‑Hermiticity directly at the kernel level.

To interpret the non‑local component equations in familiar terms, the paper adapts the Coz‑Arnold‑MacKellar current‑based localisation method. For each component j = 1,2 they consider the non‑local Schrödinger equation