Noncommutative Geometry, Semiclassical Analysis, and Weak Schatten p-Classes

The first part of the lecture will survey the main facts regarding Connes’ integration, Weyl’s laws forcompact operators and their relationships with semiclassical analysis. In particular, we will explainthe link between Connes’ integration formula and semiclassical Weyl’s laws. This will include somebackground on Schatten p-classes and the Birman-Schwinger principle. The 2nd part will presentnew 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-Zaninand Kordyukov-Sukochev-Zanin. For the Dirichlet and Neumann problems on Euclidean domainsand closed Riemannian manifolds this enables us to recover the semiclassical Weyl’s laws in thosesettings from old results of Minakshisundaram and Pleijel from the late 40s. For closed manifoldsthis also allows us to recover the celebrated Weyl’s laws of Birman-Solomyak for negative-orderpseudodifferential operators. A further set of examples is provided by Schrödinger operators associated to sub-Laplacians on sub-Riemannian manifolds, including contact manifolds and BaouendiGrushin example. Finally, we will explain how this framework enables us to get semiclassical Weyl’slaws for noncommutative tori. This will solve conjectures by Edward McDonald and the speaker.

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