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

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.