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

Abstract We study the 1D generalized Dirac oscillator with a complex Morse interaction, $f(x)=D-qe^{-\alpha x}$, in the Magueijo–Smolin (MS) and Amelino-Camelia (AC) realizations of doubly special relativity (DSR). The spatial problem factorizes into supersymmetric partners. For $D\gt 0$ and $\mathrm{Re}\, q\gt 0$, the positive metric $\eta =\exp (-\theta p/\hbar )$, $\theta =2\arg q/\alpha$, defines, on analytic bound states, a similarity transformation $\rho =\eta ^{1/2}$ to a Hermitian Morse representative. Direct Kummer analysis on the complex ray gives the real spectrum $\epsilon _n=D^2-(D-n\hbar \alpha )^2$, independent of the phase of q, with $0\le n\lt D/(\hbar \alpha )$ following from square integrability on the whole line. We relate the Morse parameters to the DSR scale k. Both DSR equations preserve the spatial spectrum but deform energy reconstruction and spinor amplitudes. On admissible branches, $E^{\rm MS}_{n,\sigma }=\sigma \Omega _n-m^2/k$ and $E^{\rm AC}_{n,\sigma }=\sigma \Omega _n+\epsilon _n/(2k)$ to first order, with $\Omega _n=\sqrt{m^2+\epsilon _n}$. On nonzero SUSY doublets, the AC generalized-eigenvalue weight is positive exactly for $\epsilon _n\lt 4k^2$. At this boundary the positive branch diverges while the negative branch remains finite, yielding a Morse excitation threshold. Quantitative use requires $|E|/k\ll 1$. The model provides an exactly solvable testbed linking pseudo-hermiticity, supersymmetry, and DSR kinematics.