PDM Klein-Gordon particles in Gödel-type Som-Raychaudhuri cosmic string spacetime background

Abstract: In G\"odel-type Som-Raychaudhuri (SR) cosmic string spacetime background, we re-cycle the Klein-Gordon (KG) oscillators and report their correct exact solutions. We argue that the mathematical collapse of the KG-equation into the 2-dimensional radial Schr\"odinger-like oscillator does not yield that the parametric characterizations of one is inherited by the other. The angular frequency (positive) in the Schr\"odinger case is replaced by an irrational frequency-like (positive and negative) in the KG-case. Inheriting the Schr\"odinger oscillator's parametric characterizations implies that at least half of the spectra (the negative part) is lost in the process. We also introduce KG-oscillators in pseudo-G\"odel SR-type spacetime that admit invariance and isospectrality with those in G\"odel SR-type spacetime background. We introduce position-dependent mass (PDM) settings to KG-particles in 4-vector and scalar Lorentz potentials in magnetic field in the G\"odel SR-type spacetime background. Four illustrative examples of fundamental nature are discussed and their exact or conditionally exact solvability are reported. Amongst are, a PDM KG-Coulombic particle in 4-vector and scalar Coulombic Lorentz potentials in magnetic field, a PDM KG-Coulombic particle in 4-vector and scalar Coulombic Lorentz potentials in magnetic field at zero vorticity, a PDM KG-Coulombic particle in equally mixed 4-vector and scalar Coulombic Lorentz potentials in magnetic field, and a quasi-free PDM KG-oscillator. We also emphasis that the biconfluent Heun polynomial approach, to the effective oscillator plus Cornell type potential, yields conditionally exact solvability that paralyzes the solution from collapsing into that of pure Coulombic one.