Semiclassical quantization and spectral limits of ħ-pseudodifferential and Berezin-Toeplitz operators

We introduce a minimalistic notion of semiclassical quantization and use it to prove that the convex hull of the semiclassical spectrum of a quantum system given by a collection of commuting operators converges to the convex hull of the spectrum of the associated classical system. This gives a quick alternative solution to the isospectrality problem for quantum toric systems. If the operators are uniformly bounded, then the convergence is uniform. Analogous results hold for non-commuting operators.