Semiclassical analysis and noncommutative geometry are distinct fields within the wider area of quantum theory. Bridges between them have been emerging recently. In this series of lectures we will explain some of these links.
In the first part of the lecture we will go over further examples of semiclassical Weyl’s laws. A first set of examples is provided by Dirac-Schroedinger operators and magnetic Schroedinger operators on open manifolds with conformally cusp metrics. Another set of examples is provided by Schr?dinger operators associated to sub-Laplacians on sub-Riemannian manifolds, including contact manifolds and Baouendi-Grushin example. The remainder of the lecture will focus on noncommutative tori. In particular, we will see how the results presented in Lecture 2 enable us to prove several conjectures made in recent joint papers with Edward McDonald. This includes semiclassical Weyl’s laws for flat and curved noncommutative tori, and a version for noncommutative tori of Birman-Solomyak’s Weyl’s law. We will further focus on the semiclassical analysis on noncommutative tori. We will present recent joint work with Edward McDonald on versions for noncommutative tori of Lieb-Thirring inequalities for negative eigenvalues of Schroedinger operators on flat and curved noncommutative tori. These inequalities imply Sobolev inequalities for noncommutativee tori. The approach is based on a version of Cwikel estimates for noncommutative tori. These estimates further allow us to extend the semiclassical Weyl’s laws to Schroedinger operators associated with $L_p$-potentials.
Raphael Ponge教授是法国人，在四川大学数学学院工作，四川省千人计划入选者。他于2000年在巴黎11大获得博士学位，师从菲尔兹奖得主Alain Connes。Ponge教授的研究领域是非交换几何，曾在美国，加拿大，日本，韩国工作，并在Adv. Math.、 J. Funct. Anal.等期刊发表了40余篇学术论文。