Title: Sharp pointwise Weyl law for Schrodinger operators with critically singular potentials

Abstract: We consider the eigenvalue problem of the Schrodinger operators with singular potentials on compact Riemannian manifolds. These singular potentials appear naturally in physics, most notably the Coulomb potential 1/|x|. The Weyl law describes the asymptotic distribution of eigenvalues, which was first proved by Hermann Weyl in 1911 and then extensively studied by numerous prominent mathematicians. The Weyl law for critically singular potentials were recently studied by Huang and Sogge. Later, Frank and Sabin considered the pointwise Weyl law in 3D, and raised the problem in other dimensions. In recent works, we completely solve this problem by a different argument. We proved the pointwise Weyl law for the Schrodinger operators with critically singular potentials, and showed that they are sharp by constructing explicit examples on flat tori. This talk is based on my joint work with Xiaoqi Huang (UMD).

About the Speaker: Cheng Zhang is a Visiting Assistant Professor at the University of Rochester.  Next year, he will join the faculty at Tsinghua University in Beijing.  He recently was awarded prestigious research support from the National Natural Science Fund for Excellent Young Scientists through the National Natural Science Foundation of China.  Prof. Zhang’s research interests include Harmonic Analysis and PDE, Spectral Geometry, Special Functions, and Applied Mathematics.  He received his Ph.D. in 2019 from Johns Hopkins University under the direction of Christopher Sogge.  Prof. Zhang is also a mentor in the Undergraduate Research Program at the University of Rochester.