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Abstract: The main goal of the paper is to address the issue of the existence ofKempf-s distortion function and the Tian-Yau-Zelditch (TYZ) asymptoticexpansion for the Kepler manifold - an important example of non compactmanfold. Motivated by the recent results for compact manifolds we constructKempf-s distortion function and derive a precise TYZ asymptotic expansion forthe Kepler manifold. We get an exact formula: finite asymptotic expansion of$n-1$ terms and exponentially small error terms uniformly with respect to thediscrete quantization parameter $m\to \infty $ and $ ho \to \infty$, $ ho$being the polar radius in $\C^n$.Moreover, the coefficents are calculated explicitly and they turned out to behomogeneous functions with respect to the polar radius in the Kepler manifold.We also prove and derive an asymptotic expansion of the obtstruction term withthe coefficients being defined by geometrical quantities. We show that ourestimates are sharp by analyzing the nonharmonic behaviour of $T m$ and theerror 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 investigatingasymptotic expansions in the framework of analytic-Gevrey spaces.

Author: Todor Gramchev, Andrea Loi

Source: https://arxiv.org/

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