Singular parametric oscillators from the one-parameter Darboux transformation of the classical harmonic oscillator

Abstract

The singular parametric oscillators obtained from the one-parameter Darboux deformation/ transformation effected upon the classical harmonic oscillator are introduced and discussed in some detail using $sin\left({\omega }_{0}t\right)$ and $cos\left({\omega }_{0}t\right)$ as seed solutions. The corresponding Ermakov–Lewis integrability problem of these parametric oscillators is also studied. It is shown that their Ermakov–Lewis invariants do not depend on the deformation parameter and are singularity-free.