  The main goal of the paper is to address the issue of the existence of
Kempf's distortion function and the Tian-Yau-Zelditch (TYZ) asymptotic
expansion for the Kepler manifold - an important example of non compact
manfold. Motivated by the recent results for compact manifolds we construct
Kempf's distortion function and derive a precise TYZ asymptotic expansion for
the Kepler manifold. We get an exact formula: finite asymptotic expansion of
$n-1$ terms and exponentially small error terms uniformly with respect to the
discrete quantization parameter $m\to \infty $ and $\rho \to \infty$, $\rho$
being the polar radius in $\C^n$.
  Moreover, the coefficents are calculated explicitly and they turned out to be
homogeneous functions with respect to the polar radius in the Kepler manifold.
We also prove and derive an asymptotic expansion of the obtstruction term with
the coefficients being defined by geometrical quantities. We show that our
estimates are sharp by analyzing the nonharmonic behaviour of $T_m$ and the
error term of the approximation of the Fubini--Study metric by $m\omega$ for
$m\to +\infty$. The arguments of the proofs combine geometrical methods,
quantization tools and functional analytic techniques for investigating
asymptotic expansions in the framework of analytic-Gevrey spaces.
