I am reading Bates and Weinstein's book 'Lectures on the Geometry of Quantization'. In Chapter 6, they defined the $\hbar$-differential operator, and showed (Theorem 6.7) that the Lagrangian submanifolds sitting inside the characteristic variety quantized to 1-st order approximate eigenfunctions to the operator. 


I am wondering if this theorem can be improved to get an asymptotic expansion of an exact solution.



Reference of $\hbar$-differential operator from symplectic geometry perspective
You can try:


* ["Spectral Asymptotics in the Semi-Classical Limit"](http://books.google.com/books/about/Spectral_Asymptotics_in_the_Semi_Classic.html?id=py3AjSvONMoC) by Dimassi and Sjostrand.
* ["An Introduction to Semiclassical and Microlocal Analysis"](https://books.google.com/books?id=44LQcwB1gusC&source=gbs_navlinks_s) by Martinez.


