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

在全局洛伦兹几何环境下分析了拉普拉斯算子和狄拉克算子的谱理论。要考虑的两种类型的洛伦兹流形是1)时间相关的，全局双曲流形和2)静态或平稳的，非全局双曲流形。研究了上述算子的自伴随扩展。除了证明了这些算子的自伴随扩展的存在性外，我们还通过分析本质自伴随性来研究这些扩展的唯一性。给出了广义相对论和量子场论在弯曲时空中的应用。求解波动方程的主要方法是应用Shubin的最新结果，证明在完全黎曼流形上，加权拉普拉斯-贝尔特拉米算子加上一个局部平方可积势在紧支撑的光滑函数空间上本质上是自伴随的。初步的研究表明，必要自伴随的充分条件是，经过适当的保角变换后，在叶理的每片叶子上诱导的黎曼度量是测地完备的。因此，本课题的第二部分是对满足此条件的时变全局双曲流形进行分类。项目的第三部分是将这些方法扩展到研究静态或平稳、非全局双曲流形中拉普拉斯算子和狄拉克算子的本质自伴随性。在最后一部分中，我们利用这些结果在解空间上构造量化所需的复杂结构。

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.