In this talk we explain a simple treatment of the quantum Birkhoff normal form for semiclassical pseudo-differential operators with smooth coefficients. The normal form is applied to describe the discrete spectrum in a generalised non-degenerate potential well, yielding uniform estimates in the energy $E$. This permits a detailed study of the spectrum in various asymptotic regions of the parameters $(E,\hslash )$, and gives improvements and new proofs for many of the results in the field. In the completely resonant...

We give an explicit formula for the symbol of a function of an operator. Given a pseudo-differential operator $\widehat{A}$ on $L^2\left({ℝ}^N\right)$ with symbol $A\in {𝒞}^{\infty }\left(T^{*}{ℝ}^N\right)$ and a smooth function $f$, we obtain the symbol of $f\left(\widehat{A}\right)$ in terms of $A$. As an application, Bohr-Sommerfeld quantization rules are explicitly calculated at order 4 in $\hslash$.