Duality between the quantum inverted harmonic oscillator and inverse square potentials

In this paper we show how the quantum mechanics of the inverted harmonic oscillator can be mapped to the quantum mechanics of a particle in a super-critical inverse square potential. We demonstrate this by relating both of these systems to the Berry-Keating system with hamiltonian $H=(xp+px)/2$. It has long been appreciated that the quantum mechanics of the inverse square potential has an ambiguity in choosing a boundary condition near the origin and we show how this ambiguity is mapped to the inverted harmonic oscillator system. Imposing a boundary condition requires specifying a distance scale where it is applied and changes to this scale come with a renormalization group (RG) evolution of the boundary condition that ensures observables do not directly depend on the scale (which is arbitrary). Physical scales instead emerge as RG invariants of this evolution. The RG flow for the inverse square potential is known to follow limit cycles describing the discrete breaking of classical scale invariance in a simple example of a quantum anomaly, and we find that limit cycles also occur for the inverted harmonic oscillator. However, unlike the inverse square potential where the continuous scaling symmetry is explicit, in the case of the inverted harmonic oscillator it is hidden and occurs because the hamiltonian is part of a larger su(1,1) spectrum generating algebra. Our map does not require the boundary condition to be self-adjoint, as can be appropriate for systems that involve the absorption or emission of particles.