Relativistic linear oscillator under the action of a constant external force. Transition amplitudes and the Green's function

We discuss an exactly solvable relativistic model of a nonrelativistic linear harmonic oscillator in the presence of a constant external force. We show that as in the nonrelativistic case, the relativistic linear oscillator in an external uniform field is unitarily equivalent to the oscillator without this field. Using two methods, we calculate transition amplitudes between energy states of the discrete spectrum of the relativistic linear oscillator under the action of a suddenly applied uniform field. We find Barut–Girardello coherent states and the Green's function in the coordinate and momentum representations. We obtain the linear and bilinear generating functions for the Meixner–Pollaczek polynomials. We prove that the relativistic wave functions, the generators of the dynamical symmetry group, and the transition amplitudes have the correct nonrelativistic limit.