Pointwise Weyl Laws for Quantum Completely Integrable Systems

TL;DR

This paper proves a new microlocalized pointwise Weyl law for the spectral functions of quantum completely integrable systems on compact manifolds, extending classical asymptotic results to joint spectral functions.

Contribution

It establishes the first microlocalized pointwise Weyl law for joint spectral functions of quantum completely integrable systems, generalizing classical results to a multivariate setting.

Findings

Derived asymptotics for joint spectral functions of QCI systems.

Extended classical Weyl law to a microlocalized, pointwise setting.

Applied results to examples including surfaces of revolution.

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 H\"ormander, following important prior contributions by G\"arding, Levitan, Avakumovi\'c, 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 $1\leq i,j\leq n$. A particularly important case is when $(M,g)$ is Riemannian and $P_1=(-\Delta)^\frac12$. We illustrate our result with several examples, including surfaces of revolution.