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 vertical bar TOO(dagger)vertical bar 0 >. The operator O represents an infinite tower of bound states with a geometric energy spectrum. Operators with higher angular momenta are briefly discussed. (C) 2011 Elsevier Inc. All rights reserved.