2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/219329This article gives 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 $(E,\h)$, and gives improvements and new proofs for many of the results in the field. In the completely resonant case we show that the pseudo-differential operator can be reduced to a Toeplitz operator on a reduced symplectic orbifold. Using this quantum reduction, new spectral asymptotics concerning the fine structure of eigenvalue clusters are proved. In the case of polynomial differential operators, a combinatorial trace formula is obtained.44 pages, 2 figuresSpectral TheoryMathematical Physics58J40, 58J50, 58K50, 47B35, 53D20text