Siegert赝态：完备性与时间演化
Siegert pseudostates: completeness and time evolution
摘要：
在Siegert赝态理论中，可以精确地计算束缚态和共振态。能量连续体被一组离散的状态所取代。散射理论中的许多问题都可以在这种形式主义的框架内解决，从而避免了处理能量连续体的需要。对于实际计算，重要的是要知道Siegert赝态的某个子集是否包含一个基。这是一个重要的问题，因为Siegert赝态具有不寻常的正交性和过完备性。使用分析和数值参数，它表明，子集的束缚态和即将离任的Siegert pseudostates形成的基础。Siegert赝态的背景下的时间演化也进行了研究。从出波格林函数的Mittag-Leffler展开式出发，导出了波包的Siegert赝态的含时展开式。在这个表达式中，使用了所有Siegert伪态--束缚态、反束缚态、传出态和传入态。每一个都以非指数的方式随时间演化。数值试验表明了该方法的准确性。
英文摘要：
Within the theory of Siegert pseudostates, it is possible to accurately calculate bound states and resonances. The energy continuum is replaced by a discrete set of states. Many questions of interest in scattering theory can be addressed within the framework of this formalism, thereby avoiding the need to treat the energy continuum. For practical calculations it is important to know whether a certain subset of Siegert pseudostates comprises a basis. This is a nontrivial issue, because of the unusual orthogonality and overcompleteness properties of Siegert pseudostates. Using analytical and numerical arguments, it is shown that the subset of bound states and outgoing Siegert pseudostates forms a basis. Time evolution in the context of Siegert pseudostates is also investigated. From the Mittag-Leffler expansion of the outgoing-wave Green's function, the time-dependent expansion of a wave packet in terms of Siegert pseudostates is derived. In this expression, all Siegert pseudostates--bound, antibound, outgoing, and incoming--are employed. Each of these evolves in time in a nonexponential fashion. Numerical tests underline the accuracy of the method.
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