  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.
