A note on scaling asymptotics for Bohr-Sommerfeld Lagrangian submanifolds

TL;DR

This paper investigates the asymptotic behavior of quantum states associated with Bohr-Sommerfeld Lagrangian submanifolds in geometric quantization, revealing a natural factorization and providing remainder estimates for their scaling asymptotics.

Contribution

It introduces a natural factorization in the scaling asymptotics of quantum states on Bohr-Sommerfeld Lagrangian submanifolds and offers precise remainder estimates.

Findings

Asymptotic concentration of quantum states on submanifolds as tensor power increases

Identification of a natural factorization in the asymptotic expansion

Provision of explicit remainder estimates for the asymptotics

Abstract

An important problem in geometric quantization is that of quantizing certain classes of Lagrangian submanifolds, so-called Bohr-Sommerfeld Lagrangian submanifolds, equipped with a smooth half-density. A procedure for this in the complex projective setting is, roughly speaking, to apply the Szego kernel of the quantizing line bundle to a certain induced delta function supported on the submanifold. If the quantizing line bundle L is replaced by its k-th tensor power, and k tends to infinity, the resulting quantum states u_k concentrate asymptotically on the submanifold. This note deals with the scaling asymptotics of the u_k's along the submanifold; in particular, we point out a natural factorization in the corresponding asymptotic expansion, and provide some remainder estimates.