This paper studies the so-called
dynamical group of the n-dimensional non-relativistic quantum mechanical Kepler
problem. This group turns out to be isomorphic to the real pseudo-orthogonal group
O(2,n + 1). First it is shown that there exists a Lie algebra 𝔾 of formal differential
operators on ℝn which has the following properties: The algebra 𝔾 is isomorphic to
the Lie algebra so(2,n + 1) and contains the formal hamiltonian operators for the
positive and negative energy spectra of the Kepler problem. This much is done
in the spirit of the work in the physics literature for the case n = 3. The
main results of the paper show that there exists a unitary representation of
the group O(2,n + 1) whose differential is a skew self adjoint extension of
the Lie algebra 𝔾. In outline, this group representation arises as a certain
intregral transform of a solution space of the (n + 1)-dimensional wave equation
on ℝn+1. The latter solution space is derived from a suitable non-unitary
induced representation of O(2,n + 1) induced by a certain maximal parabolic
subgroup.
