  A proof that minimum uncertainty states of the simplest periodic quantum
system exist in a state space that is represented by a Colombeau algebra of
generalised functions but not in Hilbert space or in the space of Schwartz
distributions is given. There are two significant generalisations of Hilbert
space that lead to the special Colombeau algebra. The first step is to a vector
space of Schwartz distributions that contains the eigenstates of operators with
continuous spectra, the second is to a Colombeau algebra that provides a
solution for a differential equation with non-constant coefficients.
  From the perspective of Colombeau algebra a rigged Hilbert space of Schwartz
distributions can be understood as a representation of the linear component of
the algebra. The utility of this representation is illustrated by a proof that
generalised eigen-decompositions exist for linear operators with periodic
continuous spectra.
