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

We study the one-dimensional generalized Dirac oscillator with a complex Morse interaction, f(x) = D − qe−αx, in the Magueijo–Smolin (MS) and Amelino-Camelia (AC) realizations of doubly special relativity (DSR). The spatial problem factorizes into supersymmetric partners. For D > 0 and Re q > 0, the positive metric η = exp ( − θp/ℏ), θ =2 q/α, defines, on analytic bound states, a similarity transformation ρ = η1/2 to a Hermitian Morse representative. Direct Kummer analysis on the complex ray gives the real spectrum εn = D2 − (D − nℏα)2, independent of the phase of q, with 0 ≤ n < D/(ℏα) 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, EMSn,σ=σ Ω ₙ-m²/k and EACn,σ=σ Ω ₙ+ε ₙ/(2k) to first order, with Ω ₙ=√m²+ε ₙ. On nonzero SUSY doublets, the AC generalized-eigenvalue weight is positive exactly for εn < 4k2. At this boundary the positive branch diverges while the negative branch remains finite, yielding a Morse excitation threshold. Quantitative use requires |E|/k ≪ 1. The model provides an exactly solvable testbed linking pseudo-Hermiticity, supersymmetry, and DSR kinematics.