Quantum Tunneling and Caustics under Inverse Square Potential

We examine the quantization of a harmonic oscillator with inverse square potential V(x)=(mω²/2) x²+g/x² on the line −∞<x<∞. We find that, for 0<g<3?²/(8m), the system admits a U(2) family of inequivalent quantizations allowing for quantum tunneling through the infinite potential barrier at x=0. These are a generalization of the conventional quantization applied to the Calogero model in which no quantum tunneling is allowed. The tunneling renders the classical caustics which arise under the potential anomalous at the quantum level, leading to the possibility of copying the profile of an arbitrary state from one side x>0, say, to the other x<0.