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Heat Kernel Expansion for Schrödinger-Type Operators on Noncompact Manifolds and Its Applications

组织者

李镛 , 唐鑫星

演讲者

颜俊榕

时间

2025年05月15日 14:00 至 15:30

地点

A3-2a-302

线上

Zoom 482 240 1589 (BIMSA)

摘要

Motivated by the study of Landau-Ginzburg models in string theory from the viewpoint of the index theorem, we explore the heat kernel expansion for Schrödinger-type operators on noncompact manifolds. Unlike in the compact case, the heat kernel in the noncompact setting exhibits new behaviors. Obtaining its precise expansion and deriving a remainder estimate requires careful analysis. We present our approach to establishing this expansion. As one key application (recent joint work with Xianzhe Dai), we study Weyl's law for such operators. In the compact case, such results follow from Karamata's Tauberian theorem, but the standard Tauberian argument does not apply in the noncompact setting. To address this, we develop a new version of Karamata's theorem. If time permits, I will further discuss how spectral invariants (such as analytic torsion and Feynman graph integrals)—for Dirac-type operators on manifolds with boundary—can be linked to Schrödinger-type operators on noncompact manifolds. This perspective suggests that spectral theory for bounded domains may sometimes be reduced to the study of Schrödinger operators on noncompact spaces, potentially leading to new techniques in studying gluing formulas for spectral invariants.