  In this paper, we study a refined L2 version of the semiclassical
approximation of projectively invariant elliptic operators with invariant Morse
type potentials on covering spaces of compact manifolds. We work on the level
of spectral projections (and not just their traces) and obtain an information
about classes of these projections in K-theory in the semiclassical limit as
the coupling constant goes to zero. An important corollary is a vanishing
theorem for the higher traces in cyclic cohomology for the spectral
projections. This result is then applied to the quantum Hall effect. We also
give a new proof that there are arbitrarily many gaps in the spectrum of the
operators under consideration in the semiclassical limit.
