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It is shown that the Stone-von Neumann theorem is inapplicable to scatteringa quantum nonrelativistic particle on a one-dimensional "short-range" potentialbarrier, since the unboundedness of the position operator plays here a crucialrole. The Shcr\"odinger representation associated with this process isreducible: long before and long after the scattering event the space of itsasymptotes represents the direct sum of the subspaces of left and rightasymptotes. 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 theabove subspaces. By the SSR any superposition of states from different coherentsectors is a mixed state, and splitting the incident wave packet into thetransmitted and reflected parts is nothing but a conversion of a pure stateinto a mixed one. The average values of any observable can be defined only forthe transmission and reflection subprocesses. The former evolves within asingle coherent sector of the Hilbert space, in the momentum representation;while the latter evolves within a single coherent sector of this space, in thecoordinate representation.

Revising the applicability of the Stone-von Neumann theorem to scattering a quantum particle on a one-dimensional potential barrier