Poisson and Szeg\"{o} kernel scaling asymptotics on Grauert tube boundaries (after Zelditch, Chang and Rabinowitz)

TL;DR

This paper reviews and extends recent results on the asymptotic behavior of Poisson and Szeg"{o} kernels on Grauert tubes, providing explicit formulas and broader rescaling ranges for applications.

Contribution

It offers explicit leading order descriptions and growth estimates for rescaled asymptotics of kernels on Grauert tubes, expanding the scope of previous work.

Findings

Explicit description of leading order terms in near-diagonal asymptotics

Growth estimates for polynomial degrees in rescaled asymptotics

Rescaling range extended to $O(\lambda^{\epsilon-1/2})$

Abstract

We review and elaborate on recent work of Chang and Rabinowitz on scaling asymptotics of Poisson and Szeg\"{o} kernels on Grauert tubes, providing additional results that may be useful in applications. In particular, focusing on the near-diagonal case, we give an explicit description of the leading order terms, and an estimate on the growth of the degree of certain polynomials describing the rescaled asymptotics. Furthermore, we allow rescaled asymptotics in a range $O\left(\lambda^{\epsilon-1/2}\right)$ in all the variables involved, where $\lambda\rightarrow+\infty$ is the asymptotic parameter, rather than rescale according to Heisenberg type.