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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., Comment: 29 pages, 7 figures

The attractive inverse square potential arises in a number of physical problems such as a dipole interacting with a charged wire, the Efimov effect, the Calgero-Sutherland model, near-horizon black hole physics and the optics of Maxwell fisheye lenses. Proper formulation of the inverse-square problem requires specification of a boundary condition (regulator) at the origin representing short-range physics not included in the inverse square potential and this generically breaks the Hamiltonian's continuous scale invariance in an elementary example of a quantum anomaly. The system's spectrum qualitatively changes at a critical value of the inverse-square coupling, and we here point out that the transition at this critical potential strength can be regarded as an example of a $\mathcal{PT}$ symmetry breaking transition. In particular, we use point particle effective field theory (PPEFT), as developed by Burgess et al [J. High Energy Phys., 2017(4):106, 2017], to characterize the renormalization group (RG) evolution of the boundary coupling under rescalings. While many studies choose boundary conditions to ensure the system is unitary, these RG methods allow us to systematically handle the richer case of nonunitary physics describing a source or sink at the origin (such as is appropriate for the charged wire or black hole applications). From this point of view the RG flow changes character at the critical inverse-square coupling, transitioning from a sub-critical regime with evolution between two real, unitary fixed points ($\mathcal{PT}$ symmetric phase) to a super-critical regime with imaginary, dissipative fixed points ($\mathcal{PT}$ symmetry broken phase) that represent perfect-sink and perfect-source boundary conditions, around which the flow executes limit-cycle evolution., Comment: 15 pages, 4 figures. This article is a contribution to the proceedings for the online seminar series on Pseudo-Hermitian Hamiltonians in Quantum Physics