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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.

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You can try:

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