A note on scaling asymptotics for Bohr-Sommerfeld Lagrangian submanifolds

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.