A Small-Scale Density of States Formula

 Let (M,g) be a C ^∞ compact Riemann manifold with classical Hamiltonian, HC ^∞ (T ^* M). Assume that the corresponding -quantization P ₁ :=Op _ (H) is quantum completely integrable. We establish an -microlocal Weyl law on short spectral intervals of size ^2−ε;∀ε>0 for various families of operators P ₁ ^u ;uI containing P ₁ , both in the mean and pointwise a.e. for uI. The -microlocalization refers to a small tubular neighbourhood of a non-degenerate, stable periodic bicharacteristic γ⊂T ^* M−0. Received: 10 December 2001 / Accepted: 23 January 2003 Published online: 2 April 2003 RID="⋆" ID="⋆" Supported in part by an Alfred P. Sloan Research Fellowship and NSERC grant OGP01720280 Communicated by P. Sarnak