Exact Quantum State for the Time-Dependent Repulsive Oscillator with a Driving Force and with a Singular Perturbation

Abstract

We derived the quantum solutions of a repulsive oscillator driven by a timedependent force and a repulsive oscillator perturbed by a singularity using unitary transformation method. The wave functions of the system perturbed by a singularity are expressed in terms of the Kummer confluent hypergeometric function while that of the driven system are expressed in terms of the Weber function. Since the repulsive oscillator is unbound, its spectrum of eigenvalues is continuous. We applied our general development to the repulsive Caldirola–Kanai oscillator.