A note on scaling asymptotics for Bohr-Sommerfeld Lagrangian submanifolds

摘要

An important problem in geometric quantization is that of quantizing certainclasses of Lagrangian submanifolds, so-called Bohr-Sommerfeld Lagrangiansubmanifolds, equipped with a smooth half-density. A procedure for this in thecomplex projective setting is, roughly speaking, to apply the Szego kernel ofthe quantizing line bundle to a certain induced delta function supported on thesubmanifold. If the quantizing line bundle L is replaced by its k-th tensorpower, and k tends to infinity, the resulting quantum states u_k concentrateasymptotically on the submanifold. This note deals with the scaling asymptoticsof the u_k's along the submanifold; in particular, we point out a naturalfactorization in the corresponding asymptotic expansion, and provide someremainder estimates.