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A time-dependent unitary (canonical) transformation is found which maps theHamiltonian for a harmonic oscillator with time-dependent real mass and realfrequency to that of a generalized harmonic oscillator with time-dependent realmass and imaginary frequency. The latter may be reduced to an ordinary harmonicoscillator by means of another unitary (canonical) transformation. A simpleanalysis of the resulting system leads to the identification of a previouslyunknown class of exactly solvable time-dependent oscillators. Furthermore, itis shown how one can apply these results to establish a canonical equivalencebetween some real and imaginary frequency oscillators. In particular it isshown that a harmonic oscillator whose frequency is constant and whose massgrows linearly in time is canonically equivalent with an oscillator whosefrequency changes from being real to imaginary and vice versa repeatedly.Comment: 7 pages, 1 figure include

Inverting a time-dependent harmonic oscillator potential by a unitary transformation and a class of exactly solvable oscillators