Nonrelativistic scale anomaly, and composite operators with complex scaling dimensions

It is demonstrated that a nonrelativistic quantum scale anomaly manifests itself in the appearance of composite operators with complex scaling dimensions. In particular, we study nonrelativistic quantum mechanics with an inverse square potential and consider a composite s-wave operator O=\psi\psi. We analytically compute the scaling dimension of this operator and determine the propagator <0|T O O^{\dagger}|0>. The operator O represents an infinite tower of bound states with a geometric energy spectrum. Operators with higher angular momenta are briefly discussed.