Construction of Exact Ermakov-Pinney Solutions and Time-Dependent Quantum Oscillators

The harmonic oscillator with a time-dependent frequency has a family of linear quantum invariants for the time-dependent Schr\"{o}dinger equation, which are determined by any two independent solutions to the classical equation of motion. Ermakov and Pinney have shown that a general solution to the time-dependent oscillator with an inverse cubic term can be expressed in terms of two independent solutions to the time-dependent oscillator. We explore the connection between linear quantum invariants and the Ermakov-Pinney solution for the time-dependent harmonic oscillator. We advance a novel method to construct Ermakov-Pinney solutions to a class of time-dependent oscillators and the wave functions for the time-dependent Schr\"{o}dinger equation. We further show that the first and the second P\"{o}schl-Teller potentials belong to a special class of exact time-dependent oscillators. A perturbation method is proposed for any slowly-varying time-dependent frequency.