Canonical transformations to action and angle variables and their representation in quantum mechanics. IV: Periodic potentials (Q1101268)

[For part III see \textit{J. Deenen}, the last author and \textit{T. H. Seligman}, ibid. 127, 458--477 (1980; Zbl 0442.70024).] A one-dimensional system of a particle in a periodic potential is considered both in classical and quantum mechanics. In the classical system, the mapping between the observables \((q,p)\) of position and momentum and the observables of angle and action is considered. An ambiguity group is described as the group of transformations that connects different points in the angle-action chart which belong to the same point \((q,p)\), and vice versa. This ambiguity group characterizes the nonbijective mapping. For periodic potentials, the ambiguity group becomes a double translation group in the position-momentum and in the angle-action charts respectively. From the classical canonical transformation the authors pass to its representation in quantum mechanics. This representation is characterized by the irreducible representations of the two translation groups. The representation corresponding to periodicity in position space yields the continuous spectrum inside the bands. It is concluded that the characteristics of the spectrum in quantum mechanics are already implicit in the classical system through the ambiguity group.