The Generalized Dirac Oscillator in Doubly Special Relativity: A Complexified Morse Interaction

We study the one-dimensional Generalized Dirac Oscillator (GDO) under Doubly Special Relativity (DSR) kinematics. The GDO extends the Dirac oscillator by replacing the linear non-minimal coupling with a general interaction function $f(x)$, thereby generating broad families of exactly solvable relativistic models and, for suitable complex choices of $f(x)$, entering the domain of $\eta$-pseudo-Hermitian and $\mathcal{PT}$-symmetric dynamics with real spectra. We present a review of the factorization (supersymmetric) structure that decouples the GDO into partner Schr\"odinger-like Hamiltonians, and we clarify how pseudo-Hermiticity and $\mathcal{PT}$ symmetry provide consistent inner products and reality conditions for the spatial spectrum. We then embed these results into two representative DSR prescriptions: the Magueijo--Smolin (MS) and the Amelino--Camelia (AC) frameworks. In this approach, the spatial problem yields a real set $\{\epsilon_n\}$, while DSR deforms the algebraic reconstruction map between $\epsilon_n$ and the relativistic energies $E_n$. The MS model induces a branch-asymmetric deformation through an energy-dependent effective mass, whereas the AC model introduces a characteristic criticality through a momentum-sector deformation, resulting in an admissibility requirement of the form $\epsilon_n<4k^2$ in the leading-order realization adopted here. As an explicit illustration, we treat a pseudo-Hermitian complexified Morse interaction, discuss the interplay between the intrinsic Morse finiteness of bound states and DSR-induced truncations, and analyze the massless limit ($m=0$), where MS collapses to the undeformed energy map while AC remains deformed.