Canonical transformations and accidental degeneracy. IV. Problems with continuous spectra

In the present series of papers we have been trying to bring out the unifying role of groups of canonical transformations in the understanding of problems of accidental degeneracy in quantum mechanics. In Paper III of this series we achieved our purposes for two-dimensional problems with discrete spectra. In the present paper we turn our attention to problems with continuous spectra. There is the well-known case of the free particle in the full plane in which the accidental degeneracy is due to the Euclidean group in two dimensions, E (2). We show that in this problem the accidental degeneracy can also be explained by an O (2,1) group of canonical transformations which provides a clue of the approach to more general problems. We also derive explicitly the group O (2,1), and not only its Lie algebra, associated with the accidental degeneracy of the Coulomb problem in two dimensions. The procedure followed in the above problems is ’’sui generis’’ and does not provide a general approach. For the latter we discuss two new problems with continuous spectra that have accidental degeneracy: the free particle in a sector of angle π/q, q integer, of the plane and the Calogero problem with continuous spectrum. For both of these problems we find the canonical transformations that map them on the free particle in the full plane. It turns out that their accidental degeneracy is explained then by the O (2,1) group of the latter problem, that we mentioned above, rather than by E (2). The procedures developed seem general enough to encompass other problems of accidental degeneracy in configuration spaces of two or more dimensions.