Pointwise Weyl laws for quantum completely integrable systems

Abstract The study of asymptotics for the spectral function of a single self-adjoint, elliptic differential or pseudodifferential operator on a compact manifold has a long history. Following important prior work, in 1968 Hörmander obtained pointwise asymptotics (or a “pointwise Weyl law”) for a single elliptic, self-adjoint pseudodifferential operator. Here, we establish a (microlocalized) pointwise Weyl law for the joint spectral functions of quantum completely integrable (QCI) systems, P ¯ = ( P 1 , P 2 , … , P n ) {\overline{P}=(P_{1},P_{2},\dots,P_{n})} . The P i {P_{i}} are first-order, classical, self-adjoint, ΨDO on a compact manifold M of dimension n . We assume that [ P i , P j ] = 0 {[P_{i},P_{j}]=0} for 1 ≤ i , j ≤ n {1\leq i,j\leq n} , that ∑ P i 2 {\sum P_{i}^{2}} is elliptic (although no single P i {P_{i}} need be elliptic), and impose a fiber-rank condition . Important examples have P 1 = ( - Δ ) 1 2 {P_{1}=(-\Delta)^{\frac{1}{2}}} on a Riemannian manifold ( M , g ) {(M,g)} with completely integrable geodesic flow. We illustrate our results with several exam