Adiabatic limits and noncommutative Weyl formula

TL;DR

This paper explores the asymptotic eigenvalue distribution of the Laplacian on foliated manifolds in the adiabatic limit, proposing a conjectural noncommutative Weyl formula inspired by semiclassical analysis.

Contribution

It introduces a new conjectural formula for eigenvalue distribution in the adiabatic limit using noncommutative geometry, extending spectral asymptotics to foliated manifolds.

Findings

Review of known results on eigenvalue asymptotics

Proposal of a noncommutative Weyl formula

Discussion on the validity of the formula in various cases

Abstract

We discuss asymptotic behavior of the eigenvalue distribution of the differential form Laplacian on a Riemannian foliated manifold when the metric on the ambient manifold is blown up in directions normal to the leaves (in the adiabatic limit). Motivated by analogies with semiclassical spectral asymptotics, we use ideas and notions of noncommutative geometry to suggest a conjectural formula for the eigenvalue distribution in the adiabatic limit, which we call noncommutative Weyl formula. We review known results and discuss the correctness of the noncommutative Weyl formula in each case.