TYZ expansion for the Kepler manifold

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 manifold. 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=1/h standing for Planck's constant and |x| tends to infinity x \in C^n. Moreover, the coefficients are calculated explicitly and they turned out to be homogeneous functions with respect to the polar radius in the Kepler manifold. We show that our estimates are sharp by analyzing the nonharmonic behaviour of T_m for m tending to infinity.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.