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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 cyclicdoi:10.1515/crll.2005.2005.581.193 fatcat:rucx62gujrautcfvrjqeufxequ