  It is shown that the Stone-von Neumann theorem is inapplicable to scattering
a quantum nonrelativistic particle on a one-dimensional "short-range" potential
barrier, since the unboundedness of the position operator plays here a crucial
role. The Shcr\"odinger representation associated with this process is
reducible: long before and long after the scattering event the space of its
asymptotes represents the direct sum of the subspaces of left and right
asymptotes. There is a dichotomous-context-induced superselection rule (SSR),
in which the role of a superselection operator is played by the Pauli matrix
$\sigma_3$ and the role of superselection (coherent) sectors is played by the
above subspaces. By the SSR any superposition of states from different coherent
sectors is a mixed state, and splitting the incident wave packet into the
transmitted and reflected parts is nothing but a conversion of a pure state
into a mixed one. The average values of any observable can be defined only for
the transmission and reflection subprocesses. The former evolves within a
single coherent sector of the Hilbert space, in the momentum representation;
while the latter evolves within a single coherent sector of this space, in the
coordinate representation.
