Abstract
An Ermakov system consists of a pair of coupled non-linear differential equations which share a joint constant of motion named Ermakov invariant. One of those equations, non-linear, is frequently referred to as the Ermakov-Pinney equation; the other equation may be thought of as describing a dynamical system: a harmonic oscillator with time-dependent frequency. In this paper, we revise the Quantum Arnold Transformation, a unitary operator mapping the solutions of the Schrödinger equation for time-dependent (even damped) harmonic oscillators, described by the Generalized Caldirola-Kanai equation, into solutions for the free particle. With this tool, we elucidate the existence of Ermakov-type invariants in classically linear systems at the classical and quantum levels. We also provide more general Ermakov-type systems and the corresponding invariants, together with a physical interpretation.
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