Pointwise Weyl Laws for Quantum Completely Integrable Systems

Abstract

The study of the asymptotics of the spectral function for self-adjoint, elliptic differential, or more generally pseudodifferential, operators on a compact manifold has a long history. The seminal 1968 paper of Hörmander, following important prior contributions by Gärding, Levitan, Avakumović, and Agmon-Kannai (to name only some), obtained pointwise asymptotics (or a "pointwise Weyl law") for a single elliptic, self-adjoint operator. Here, we establish a microlocalized pointwise Weyl law for the joint spectral functions of quantum completely integrable (QCI) systems, $\overline{P}=(P_1,P_2,\dots ,P_n)$, where $P_i$ are first-order, classical, self-adjoint, pseudodifferential operators on a compact manifold $M^n$, with $\sum P_i^2$ elliptic and $[P_i,P_j]=0$ for $1\le i,j\le n$. A particularly important case is when $(M,g)$ is Riemannian and $P_1=(-\hat{I}”{)}^{\frac{1}{2}}$. We illustrate our result with several examples, including surfaces of revolution.

32 pages