In the second lecture, I will show how to develop a full microlocal theory: how to localize quantum mechanics in phase space? Using the notion of semiclassical wavefront set, we obtain a very flexible calculus that can be used to prove Bohr-Sommerfeld rules at any order for a very accurate description of the spectrum of semiclassical operators. I will also give a quick overview of what can be done for non selfadjoint operators using Sjöstrand's theory of complexified phase spaces.

References

• Terry Tao blog on Weyl quantization
• S. Vũ Ngọc, Symplectic Techniques for Semiclassical Completely Integrable Systems, 2005
• S. Vũ Ngọc, Systèmes intégrables semi-classiques: du local au global, 2006
• D. Sepe and S. Vũ Ngọc, Integrable systems, symmetries, and quantization, 2017