The Generalized Klein--Gordon Oscillator in Doubly Special Relativity: A Complexified Morse Interaction

We investigate the one-dimensional \emph{Generalized Klein--Gordon Oscillator} (G-KGO) within Doubly Special Relativity (DSR) kinematics. The G-KGO extends the Klein--Gordon oscillator by replacing the usual linear non-minimal coupling with a general interaction function $f(x)$, leading to a factorized (SUSY-like) Schr\"odinger operator $H_-=\mathcal{A}^{\#}\mathcal{A}$ whose real spatial spectrum $\{\epsilon_n\}$ can be ensured either by Hermiticity or, for complex $f(x)$, by $\eta$-pseudo-Hermiticity and/or $\mathcal{PT}$ symmetry with a consistent metric inner product \cite{Bender1998,Bender2007RPP,BBJ2002PRL,Mostafazadeh2002,Mostafazadeh2003,Mostafazadeh2010,ElGanainy2018NatPhys}. DSR is then implemented at the level of the energy reconstruction map $\epsilon_n\mapsto E_{n,\pm}$, and we provide closed-form Magueijo--Smolin (MS) and Amelino--Camelia (AC) branches.