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We treat the quantum dynamics of a harmonic oscillator as well as itsinverted counterpart in the Schr\"odinger picture. Generally in the most papersof the literature, the inverted harmonic oscillator is formally obtained fromthe harmonic oscillator by the replacement of {\omega} to i{\omega}, this leadsto unbounded eigenvectors. This explicitly demonstrates that there are someunclear points involved in redefining the variables in the harmonic oscillatorinversion. To remedy this situation, we introduce a scaling operator (Dysontransformation) by connecting the inverted harmonic oscillator to ananti-PT-symmetric harmonic oscillator, we obtain the standard quasi-Hermiticityrelation which would ensure the time invariance of the eigenfunction's norm. Wegive a complete description for the eigenproblem. We show that thewavefunctions for this system are normalized in the sense of the pseudo-scalarproduct. A Gaussian wave packet of the inverted oscillator is investigated byusing the ladder operators method. This wave packet is found to be associatedwith the generalized coherent state that can be crucially utilized forinvestigating the mean values of the space and momentum operators. We find thatthese mean values reproduce the classical motion.

Anti-PT-symmetric harmonic oscillator and its relation to the inverted harmonic oscillator