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Using an abstract notion of semiclassical quantization for self-adjointoperators, we prove that the joint spectrum of a collection of commutingsemiclassical self-adjoint operators converges to the classical spectrum givenby the joint image of the principal symbols, in the semiclassical limit. Thisincludes Berezin-Toeplitz quantization and certain cases of$\hbar$-pseudodifferential quantization, for instance when the symbols areuniformly bounded, and extends a result by L. Polterovich and the authors. Inthe last part of the paper we review the recent solution to the inverse problemfor quantum integrable systems with periodic Hamiltonians, and explain how italso follows from the main result in this paper.

Spectral limits of semiclassical commuting self-adjoint operators