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

TL;DR

This paper explores a generalized relativistic quantum oscillator model within Doubly Special Relativity, revealing new solvable models with complex interactions, and analyzing their spectral properties and physical implications.

Contribution

It introduces a broad class of exactly solvable Dirac oscillator models with complex interactions under DSR, analyzing their spectral and symmetry properties within different DSR frameworks.

Findings

Identified real spectra for complexified Morse interactions in DSR models.

Demonstrated how DSR deforms the energy spectrum and algebraic relations.

Analyzed the massless limit and its impact on the models' behavior.

Abstract

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.