Title: The Klein-Gordon equation on asymptotically Minkowski spacetimes: causal propagators Show Chinese title

Title: 渐近闵可夫斯基时空上的克莱因-戈登方程：因果传播子

Abstract: We construct the causal (forward/backward) propagators for the massive Klein-Gordon equation perturbed by a first order operator which decays in space but not necessarily in time. In particular, we obtain global estimates for forward/backward solutions to the inhomogeneous, perturbed Klein-Gordon equation, including in the presence of bound states of the limiting spatial Hamiltonians. To this end, we prove propagation of singularities estimates in all regions of infinity (spatial, null, and causal) and use the estimates to prove that the Klein-Gordon operator is an invertible mapping between adapted weighted Sobolev spaces. This builds off work of Vasy in which inverses of hyperbolic PDEs are obtained via construction of a Fredholm mapping problem using radial points propagation estimates. To deal with the presence of a perturbation which persists in time, we employ a class of pseudodifferential operators first explored in Vasy's many-body work. Show Chinese abstract

Abstract: 我们构建了因果（前向/后向）传播子，用于描述由一个空间衰减但不一定在时间上衰减的一阶算子扰动的有质量 Klein-Gordon 方程。特别是，我们获得了非齐次、扰动 Klein-Gordon 方程前向/后向解的整体估计，包括在极限空间哈密顿量存在束缚态的情况下。为此，我们在无穷远的所有区域（空间、零和因果）证明了奇点传播的估计，并利用这些估计证明了 Klein-Gordon 算子是在自适应加权Sobolev空间之间的一个可逆映射。这项工作建立在Vasy的研究基础上，他通过构造径向点传播估计的Fredholm映射问题来获得双曲偏微分方程的逆。为了处理持久存在的扰动，我们采用了Vasy在多体工作中首次探讨的一类伪微分算子。