Self-adjointness of Laplace and Dirac operators on Lorentzian manifolds foliated by noncompact hypersurfaces

非紧超曲面洛伦兹流形上拉普拉斯和狄拉克算子的自伴性

• 批准号: 441840529
• 负责人: Professor Dr. Felix Finster
• 金额: --
• 依托单位: Lehrstuhl für Mathematik
• 依托单位国家: 德国
• 项目类别: Priority Programmes
• 财政年份: 2020
• 资助国家: 德国
• 起止时间: 2019-12-31 至 2023-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/441840529?language=en
• 关键词: Self; adjointness; Laplace; Dirac; operators

The spectral theory of Laplace(-Beltrami) and Dirac operators is analyzed in a global Lorentzian geometric setting. The two types of Lorentzian manifolds to be considered are 1) Time dependent, globally hyperbolic manifolds and 2) Static or stationary, non-globally hyperbolic manifolds. Self-adjoint extensions of the above operators are studied. In addition to proving the existence of self-adjoint extensions of those operators, we study the uniqueness of those extensions by analyzing essential self-adjointness. Applications to General Relativity and quantum field theory in curved spacetime are worked out.The main method for the wave equation is to apply recent results by Shubin showing that on a complete Riemannian manifold, the weighted Laplace-Beltrami operator plus a locally square integrable potential is essentially self-adjoint on the space of smooth functions of compact support. Preliminary works show that a sufficient condition for essential self-adjointness is that, after a suitable conformal transformation, the induced Riemannian metric is geodesically complete on each leaf of the foliation. Consequently, the second part of this project is to classify the time-dependent globally hyperbolic manifolds for which this condition can be satisfied. The third part of the project is to extend these methods to the study of essential self-adjointness of the Laplace and Dirac operators in static or stationary, non globally hyperbolic manifolds. In the last part we use these results to construct complex structures on the solution spaces as needed for the quantization.

在整体洛伦兹几何条件下，分析了拉普拉斯（-Beltrami）算子和Dirac算子的谱理论.要考虑的两种类型的洛伦兹流形是1）依赖于时间的全局双曲流形和2）静态或静止的非全局双曲流形。研究了上述算子的自伴扩张。除了证明这些算子的自伴扩张的存在性之外，我们还通过分析本质自伴性来研究这些扩张的唯一性。在弯曲时空中应用广义相对论和量子场论.波动方程的主要方法是应用Shubin最近的结果，即在完备的黎曼流形上，加权Laplace-Beltrami算子加上局部平方可积势在紧支集的光滑函数空间上本质上是自伴的.初步的工作表明，本质自伴的一个充分条件是，经过适当的保角变换，诱导黎曼度量是测地线上完成的每一个叶子的叶理。因此，这个项目的第二部分是分类的时间依赖的整体双曲流形，这一条件可以满足。该项目的第三部分是将这些方法扩展到静态或平稳的非全局双曲流形中的拉普拉斯和狄拉克算子的本质自伴性的研究。在最后一部分中，我们使用这些结果来构建复杂的结构的解决方案空间所需的量化。