Title: Virtual states and generalized completeness relation in the Friedrichs Model Show Chinese title

Title: 虚拟态和弗里德里希斯模型中的广义完备性关系

Abstract: We study the well-known Friedrichs model, in which a discrete state is coupled to a continuum state. By examining the pole behaviors of the Friedrichs model in a specific form factor thoroughly, we find that, in general, when the bare discrete state is below the threshold of the continuum state, there should also be a virtual-state pole accompanying the bound-state pole originating from the bare discrete state as the coupling is turned on. There are also other second-sheet poles originating from the singularities of the form factor. We give a general argument for the existence of these two kinds of states. As the coupling is increased to a certain value, the second-sheet poles may merge and become higher-order poles. We then discuss the completeness relations incorporating bound states, virtual states, and resonant states corresponding to higher-order poles. Show Chinese abstract

Abstract: 我们研究了著名的弗里德里希斯模型，在该模型中，一个离散态与连续态耦合。 通过彻底检查弗里德里希斯模型在特定形式因子下的极点行为，我们发现，一般来说，当裸离散态位于连续态的阈值以下时，当耦合开启时，应该还有一个虚态极点伴随着来自裸离散态的束缚态极点。 还有其他来自形式因子奇点的第二叶极点。 我们给出了这两种状态存在的一般性论证。 当耦合增加到某个值时，第二叶极点可能会合并并成为高阶极点。 然后我们讨论了包含对应于高阶极点的束缚态、虚态和共振态的完备性关系。