The authors considers the derivation of some statistical results for the 'inverted' harmonic oscillator (one with negative kinetic and potential energies), which are equivalent to ones that exist for the more familiar simple harmonic oscillator. It is an object that is frequently employed in modelling amplifiers in quantum optics, but also arises in statistical and quantum mechanics and is of interest in its own right. The author hopes that this account may help to elucidate its role as a potential amplifier.
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Sam Tarzi (1988) studied this question.