Spectral theory with non-unitary twists

黎曼流形的谱理论是一个重要的数学领域，对许多其他数学领域（如表示论、数论、谐波分析和数学物理）都有很大的影响。在过去的几年里，人们对非紧空间的谱理论（特别是所谓的开系统）以及黎曼局部对称空间的谱理论产生了极大的兴趣。本课题将研究非紧黎曼局部对称空间的谱性质。这类扭曲远远超出了一组在尖端是酉的表示，它构成了一个边界，直到我们目前可能期望一个扭曲谱理论。我们的研究将集中在双曲空间（因此，非紧型和秩1的黎曼局部对称空间），但我们也将把预期结果扩展到更高秩的局部对称空间以及具有双曲末端的非局部对称空间。我们将研究几何和动力学在不同类型的末端如何控制这些空间的共振，固有特征值，特征函数等的性质。我们将进一步研究这些谱目标的渐近性质。换句话说，我们将从两个方面来理解“无穷几何”的主旨。该项目在研究兴趣、考虑的空间和应用的技术方面与SPP 2026中的其他项目密切相关。

The spectral theory of Riemannian manifolds is an important area of mathematics with strong influence on many other fields of mathematics (e.g., representation theory, number theory, harmonic analysis, and mathematical physics). During the last few years, a huge interest was risen in spectral theory of noncompact spaces (in particular of so-called open systems) as well as in a spectral theory for Riemannian locally symmetric spaces with non-unitary twists.With this project we will study the spectral properties of noncompact Riemannian locally symmetric spaces with twists of non-expanding cusp monodromy. This class of twists reaches far beyond the set of representations that are unitary at cusps, and it constitutes a frontier until which we might currently expect a twisted spectral theory. Our investigations will focus on hyperbolic spaces (thus, Riemannian locally symmetric spaces of noncompact type and of rank 1) but we will also carry out first steps towards an extension of the expected results to locally symmetric spaces of higher rank as well as to non locally symmetric spaces with hyperbolic ends. We will investigate how the geometry and dynamics at the various types of ends govern the properties of the resonances, proper eigenvalues, eigenfunctions, etc. of these spaces. Further we will study the asymptotic properties of these (sets of) spectral objects. In order words, we will understand here the leitmotiv "geometry at infinity" in two ways.This project is closely related to other projects in the SPP 2026 with regard to research interests, considered spaces and techniques applied.