The logarithmic phase singularity in the inverted harmonic oscillator

A spacetime singularity is located at the center of a black hole and surrounded by an event horizon, separating spacetime into two disjunct regions: one of them accessible to an outside observer and one that is not. At the event horizon, a logarithmic phase singularity emerges in the mode functions of a massless scalar field, being characteristic for Hawking radiation emitted by the black hole. We demonstrate that related features are present in the elementary quantum system of an inverted harmonic oscillator. Central to our analysis are the energy eigenfunctions of this system and their phase space representation. At first glance, neither a horizon nor a logarithmic phase dependence are apparent. However, both features are hidden in phase space and revealed by a suitable coordinate transformation. In particular, we show that the Fourier transform of a logarithmic phase leads to an expression that is reminiscent of a specific quantum statistics, governing the reflection and transmission coefficients of the inverted harmonic oscillator.