The Bargmann transform and regularization of the 2, 3, 5-dimensional Kepler problem

Abstract

We introduce a Bargmann transform for the space L2(Sn) of square integrable functions on the n=2,3,5 dimensional unit sphere Sn inmersed in Rn+1. This is done on base of the Hopf fibration for the spheres Sk↦Sd with (k,d)=(1,1),(3,2),(7,4) and a suitable canonical transformation relating two different ways to regularize the n=2,3,5 dimensional Kepler problem (with fixed negative energy) involving the null complex quadric Qn inmersed in Cn+1. We prove the unitarity of the Bargmann transform onto a suitable space of analytical functions. We give reproducing kernels for these spaces of analytical functions which, for the cases n=3,5, are defined as the kernel of quantizations of restrictions for regularizations of the classical Kepler problem. We give an inversion formula for our Bargmann transform. We also give a set of coherent states for L2(Sn), n=2,3,5 and their semiclassical asymptotics (ℏ→0). Our Bargmann transform is actually a coherent states transform. Additionally, we use the moment map technique in order to construct a map with values in Qn that gives the base to define our canonical transformation.