We construct a nonunitary transformation that relates a given "asymptotically free" conformal quantum mechanical system H f with its confined, harmonically trapped version H c . In our construction, Jordan states corresponding to the zero eigenvalue of H f , as well as its eigenstates and Gaussian packets, are mapped into the eigenstates, coherent states, and squeezed states of H c , respectively. The transformation is an automorphism of the conformal slð2; RÞ algebra of the nature of the

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... -order root of the identity transformation, to which a complex canonical transformation corresponds on the classical level being the fourth-order root of the spatial reflection. We investigate the one-and two-dimensional examples that reveal, in particular, a curious relation between the two-dimensional free particle and the Landau problem.