The Hilbert space is the space which is usually chosen as the space of state vectors. In addition, the operators of quantum mechanics act on that space. However, the Hilbert space cannot provide a proper mathematical structure to define Dirac formulation. In particular, the use of Dirac formalism on the domain of definition of an observable leads to some physical contradictions. One example arises from the Infinite Potential Well System (IPWS) which is one of the most fundamental systems of quantum mechanics. Our aim in this paper is the explicit construction of the Gel'fand triplet for the IPWS.

中文翻译：

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无限势井系统Gel'fand三重态空间结构的构建

希尔伯特空间是通常选择作为状态向量空间的空间。此外，量子力学的算子作用于该空间。然而，希尔伯特空间无法提供适当的数学结构来定义狄拉克公式。特别是，在可观测量的定义领域使用狄拉克形式主义会导致一些物理矛盾。无限势阱系统（IPWS）就是一个例子，它是量子力学最基本的系统之一。我们本文的目的是为 IPWS 显式构造 Gel'fand 三元组。