A small-scale density of states formula (Q1416899)

Let \(( M,g) \) be a compact, two-dimensional Riemannian manifold and \(H\in C^{\infty }( T^{\ast }M) \) an usual Hamiltonian. Let \(P_{1}=Op_{\hbar }( H) \) be the corresponding self-adjoint \(\hbar \)-quantization where \(Op_{\hbar }( a) \) denotes the semiclassical Weyl pseudodifferential operator quantizing \(a\). Assume that \(P_{1}\) is quantum completely integrable i.e. there exists \(P_{2}=Op_{\hbar }( p_{2}) \) such that \([ P_{1},P_{2}] =0\). The main result of this paper, namely theorem 0.5 from the introduction, gives a \(\hbar \)-microlocal Weyl law on short spectral intervals of length \(h^{2-\varepsilon }\) for any \(\varepsilon >0\) and for various families of operators \(P_{1}^{u},u\in I=[ 1-\varepsilon ,1+\varepsilon ] \) containing \(P_{1}\), both in the mean and pointwise a.e. \(u\in I\).