Robin边界条件下全局双曲时空上的Hadamard参数解：基本解与Hadamard态
The Hadamard parametrix on globally hyperbolic spacetimes with Robin boundary conditions: Fundamental solutions and Hadamard states
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中文总结 AI 辅助
本文在带类时边界的全局双曲时空中，证明了Robin边界条件下Klein-Gordon方程基本解的存在唯一性，并构造了满足Hadamard形式的Robin-Hadamard态，推广了Radzikowski定理。
中文摘要 AI 辅助
在一个$d$维（$d\geq 2$）且具有类时边界$(\mathcal{M},g)$的全局双曲时空中，我们研究了带有Robin边界条件的Klein-Gordon方程。首先，我们证明了超前和推迟基本解的存在性和唯一性，并进一步讨论了它们的结构性质。其次，在$\partial\mathcal{M}$的无穷小凸性和破碎零双特征线的边界反射局部有限性的假设下，我们利用Melrose和Sjöstrand的奇性传播定理，刻画了它们在内部$\mathring{\mathcal{M}}$中的波前集。在论文的第二部分，我们引入了Robin-Hadamard两点关联函数的概念，将其定义为$\mathcal{M}\times\mathcal{M}$上的一个正分布，具有指定的反对称部分和奇性结构，后者通过其波前集来表征。我们通过一个局部Hadamard形式来补充这一定义。在相差一个光滑余项的范围内，该形式由标准的定向Hadamard参数解以及与被反射的零光线相关的额外贡献组成。它们的系数由输运方程决定，并辅以边界处的Robin匹配条件。该构造适用于任意但有限次数的反射。我们证明了这一局部表述与波前集条件等价，从而在存在类时边界的情况下建立了Radzikowski定理的对应物。最后，从合适的静态背景上的基态出发，并利用形变论证，我们证明了在所考虑的时空类上Robin-Hadamard两点关联函数的存在性。
英文摘要
On a $d$-dimensional, $d\geq 2$, globally hyperbolic spacetime with a timelike boundary $(\mathcal{M},g)$, we investigate the Klein-Gordon equation with Robin boundary conditions. First of all, we prove existence and uniqueness of the advanced and retarded fundamental solutions, discussing in addition their structural properties. Secondly, under the hypothesis of infinitesimal convexity of $\partial\mathcal{M}$ and of local finiteness of boundary reflections for broken null bicharacteristics, we characterize their wavefront set in the interior $\mathring{\mathcal{M}}$, using the propagation of singularities theorem of Melrose and Sjöstrand. In the second part of the paper, we introduce a notion of Robin-Hadamard two-point correlation function as a positive distribution on $\mathcal{M}\times\mathcal{M}$ with prescribed antisymmetric part and singular structure, the latter being characterized in terms of its wavefront set. We complement this definition with a local Hadamard form. Up to a smooth remainder, this consists of the standard directed Hadamard parametrix together with additional contributions associated with the reflected null rays. Their coefficients are determined by transport equations supplemented by Robin matching conditions at the boundary. The construction applies to an arbitrary, but finite number of reflections. We prove that this local formulation is equivalent to the wavefront set condition, thereby establishing a counterpart of Radzikowski's theorem in the presence of a timelike boundary. Finally, starting from the ground state on suitable static backgrounds and using a deformation argument, we prove existence of Robin-Hadamard two-point correlation functions on the class of spacetimes considered.
发表机构
- Università degli Studi di Pavia（帕维亚大学）
- INFN, Sezione di Pavia（意大利国家核物理研究所帕维亚分部）
- INdAM, Sezione di Pavia（意大利国家高等数学研究所帕维亚分部）
- University of York（约克大学）
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