On Scale Invariance and Anomalies in Quantum Mechanics

Abstract

We re-consider the quantum mechanics of scale invariant potentials in two dimensions. The breaking of scale invariance by quantum effects is analyzed by the explicit evaluation of the phase shift and the self-adjoint extension method. We argue that the breaking of scale invariance reported in the literature for the $\delta$(r) potential, is an example of explicit and not an anomaly or quantum mechanical symmetry breaking.