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 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.