Adiabatic limits and noncommutative Weyl formula

We discuss asymptotic behavior of the eigenvalue distribution of the differential form Laplacian on a Riemannian foliated manifold when the metric on the ambient manifold is blown up in directions normal to the leaves (in the adiabatic limit). Motivated by analogies with semiclassical spectral asymptotics, we use ideas and notions of noncommutative geometry to suggest a conjectural formula for the eigenvalue distribution in the adiabatic limit, which we call noncommutative Weyl formula. We review known results and discuss the correctness of the noncommutative Weyl formula in each case.