北航数学论坛学术报告

--- 分析、偏微分方程与动力系统讨论班(2025春季第16讲)

Spectral distribution for the twisted Laplacian on hyperbolic surfaces

金龙（清华大学）

时间：2025年06月23日（周一）下午15:00-16:00

地点：沙河主楼 E806

摘要：In this talk, we discuss the spectrum of the twisted Laplacian operator on a compact hyperbolic surface. The twisted Laplacian associated with a harmonic form is obtained by conjugating the usual Laplacian-Beltrami operator by an integral of the harmonic form. It is also the Bochner Laplacian associated with the corresponding one-dimensional representation. When the harmonic form is real, the twisted Laplacian is non-self-adjoint but still has discrete spectrum in the complex plane. We will review its spectral theory and connection with the twisted Selberg zeta function. In particular, we show that although most of the spectrum is concentrated near the real axis, there are infinite many eigenvalues away from the real axis, at least when the harmonic form is large enough. This implies the failure of the asymptotic version of Riemann hypotheses for the twisted Selberg zeta function as well as the failure of quantum unique ergodicity. This is joint work with Gong Yulin.

报告人简介: 金龙，现任清华大学数学中心副教授。2010年本科毕业于北京大学，2015年博士毕业于加州大学伯克利分校，导师为Maciej Zworski。2015-2018年先后在哈佛大学和普度大学博士后工作。2018年起在清华大学工作。研究领域为微局部分析，谱理论和散射理论。主要工作发表于Acta Math.，Journal of AMS, Math. Ann., Comm. Math. Phys., Trans. AMS, Analysis & PDE等。

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