An abstract stationary approach to perturbation of continuous spectra and scattering theory
An abstract stationary approach to perturbation of continuous spectra and scattering theory
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连续光谱扰动的抽象平稳方法和散射理论
DOI：
10.1007/bf02786670
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发表时间：
1967
期刊：
影响因子：
--
通讯作者：
S. Kuroda
中科院分区：
文献类型：
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作者：
S. Kuroda
本文的主要目的是发展一种新的方法来研究连续光谱和散射的微扰理论。具体而言，它表明，已分别处理由不同的方法，可以系统地处理由一个相同的方法的几个问题。作为必要的工具之一，我们还制定了可分希尔伯特空间中的自伴算子的谱表示定理。这似乎给出了一种新的表示，这样的一个运营商的乘法运营商在一定的空间的向量值函数。本文的大部分结果在作者的一篇简要文章中以粗略的形式公布[16]，其中还包含对背景的解释和对我们方法的启发性描述。因此，本文专门致力于一般配方和详细的证明。在这一领域的最新研究方法可以大致分为两类。第一种方法是平稳方法，它在光滑摄动问题中是最有效的。在这里，”光滑扰动”是在一个模糊的意义上使用的，它的意思是Friedrichs [9]和Rejto [18]的温和扰动，Ikebe [10]关于Schr 6dinger算子的工作，Faddeev [8]关于Friedrichs模型的工作，等等。在这种方法中，问题通常是在特定的谱表示空间中解决的。相反，在具有迹条件的摄动问题中，第二种方法，即与时间有关的方法，
The main purpose of the present paper is to develop a new approach to the theory of perturbation of continuous spectra and scattering. Specifically, it is shown that several problems which have been treated separately by different methods can be handled systematically by one and the same approach. As one of the necessary tools we also formulate a spectral representation theorem for a self-adjoint operator in a separable Hilbert space. This gives seemingly a new kind of representation of such an operator by a multiplicative operator in a certain space of vector valued functions. Most of the results of the present paper was announced in a rough form in an expository article of the writer [16], which also contains an explanation of the background and a heuristic account of our approach. Thus, the present paper is devoted exclusively to a general formulation and a detailed proof. The methods of recent investigation in this field may be classified, roughly speaking, into two types. The first one, the stationary method, has been most effective in the problems of smooth perturbation. Here," smooth perturbation" is used in a vague sense and what is meant by it is gentle perturbation of Friedrichs [9] and Rejto [18], a work of Ikebe [10] on Schr6dinger operators, a work of Faddeev [8] on the Friedrichs' model, and so on. In this method a problem is usually solved in a certain spectral representation space suitably chosen in each individual case. On the contrary, in the problems of perturbation with a trace condition the second method, the time-dependent one, has been