  We settle affirmatively the isospectral problem for quantum toric integrable
systems: the semiclassical joint spectrum of such a system, given by a sequence
of commuting Toeplitz operators on a sequence of Hilbert spaces, determines the
classical integrable system given by the symplectic manifold and Poisson
commuting functions, up to symplectomorphisms. We also give a full description
of the semiclassical spectral theory of quantum toric integrable systems. This
type of problem belongs to the realm of classical questions in spectral theory
going back to pioneer works of Colin de Verdiere, Guillemin, Sternberg and
others in the 1970s and 1980s.
