TYZ expansion for the Kepler manifold

TL;DR

This paper establishes the existence and explicit form of the TYZ asymptotic expansion for the non-compact Kepler manifold, including precise coefficients and error estimates, extending results known for compact manifolds.

Contribution

It constructs Kempf's distortion function and derives a finite asymptotic expansion for the Kepler manifold, with explicit coefficients and sharp error bounds, a novel extension to non-compact settings.

Findings

Finite asymptotic expansion with n-1 terms

Explicit calculation of coefficients as homogeneous functions

Sharp estimates of the error terms and obstruction expansion

Abstract

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.