Quasimodes and Bohr-Sommerfeld Conditions for the Toeplitz Operators

Abstract

This article is devoted to the quantization of the Lagrangian submanifolds in the context of geometric quantization. The objects we define are similar to the Lagrangian distributions of the cotangent phase space theory. We apply this to construct quasimodes for the Toeplitz operators and we state the Bohr-Sommerfeld conditions under the usual regularity assumption. To compare with the Bohr-Sommerfeld conditions for a pseudodifferential operator with small parameter, the Maslov index, defined from the vertical polarization, is replaced with a curvature integral, defined from the complex polarization. We also consider the quantization of the symplectomorphisms, the realization of semi-classical equivalence between two different quantizations of a symplectic manifold and the microlocal equivalences.