| Résume | In this talk, I will present new results regarding semiclassical Weyl’s laws and integration formulas for noncommutative manifolds (i.e., spectral triples). This improves and simplifies recent results of McDonald-Sukochev-Zanin and Kordyukov-Sukochev-Zanin. For the Dirichlet and Neumann problems on Euclidean domains and closed Riemannian manifolds this enables us to recover the semiclassical Weyl’s laws in those settings from old results of Minakshisundaram and Pleijel from the late 40s. For closed manifolds this also allows us to recover the celebrated Weyl’s laws of Birman-Solomyak for negative-order pseudodifferential operators. A further set of examples is provided by Schrödinger operators associated to sub-Laplacians on sub-Riemannian manifolds, including contact manifolds and the Baouendi-Grushin example. Finally, we will explain how this framework enables us to get semiclassical Weyl’s laws for noncommutative tori. This will solve conjectures by Edward McDonald and myself. |
