Nonlocal Generalized Dirac Oscillators in (1 + 1) Dimensions

TL;DR

This paper introduces a nonlocal extension of the generalized Dirac oscillator in (1+1) dimensions, deriving explicit kernels, pseudo-Hermiticity conditions, and local interpretations, supported by analytical benchmarks and a finite-rank model.

Contribution

It develops a nonlocal Dirac oscillator formalism with explicit kernel expressions, pseudo-Hermiticity criteria, and a method to interpret nonlocal effects locally, including analytical solutions and a separable model.

Findings

Derived explicit supersymmetric partner kernels.

Established a kernel-level pseudo-Hermiticity condition.

Provided analytical benchmarks and a finite-rank separable model.

Abstract

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 Schr\"odinger-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 $\eta=e^{-\theta p_x}$ leads to a simple sufficient \emph{kernel-level} pseudo-Hermiticity constraint, $f(x+\ii\hbar\theta,x'+\ii\hbar\theta)=f^*(x',x)$, 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 Schr\"odinger 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.