  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
$\mathfrak{sl}(2,{\mathbb R})$ algebra of the nature of the fourth-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.
