Continuous‐Representation Theory. IV. Structure of a Class of Function Spaces Arising from Quantum Mechanics
Continuous‐Representation Theory. IV. Structure of a Class of Function Spaces Arising from Quantum Mechanics
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量子力学中的一类函数空间的连续表示理论。
DOI：
10.1063/1.1704190
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发表时间：
1964
期刊：
影响因子：
--
通讯作者：
J. Klauder
中科院分区：
文献类型：
--
作者：
J. McKenna;J. Klauder
提出了通过有界、连续、多维相空间函数ψ(p,q)对希尔伯特空间的连续表示的严格发展。结果表明，这些函数形成了 L2(p, q) 的闭子空间，其元素是函数而不是等价类。研究了微分性质，并指出存在多种定义，其中 ψ(p, q) 具有所有阶的连续导数。在这些定义之一中，每个 ψ(p, q) 与多维完整函数 f(q − ip) 成比例，从而在整个函数的巴格曼希尔伯特空间与连续表示的一个示例之间建立了联系。我们的注意力集中在通过再现内核作为 Aronszajn 一般理论的特例的连续表示的纯功能表征上。连续表示中各种运算符的属性都经过仔细定义。
A rigorous development of the continuous representation of Hilbert space by bounded, continuous, multidimensional phase‐space functions ψ(p, q) is presented. It is shown that these functions form a closed subspace of L2(p, q) whose elements are functions and not equivalence classes. Differential properties are investigated and it is pointed out that there are a multitude of definitions whereby ψ(p, q) possesses continuous derivatives of all orders. In one of these definitions, each ψ(p, q) is proportional to a multidimensional, entire function f(q − ip), establishing a connection between Bargmann's Hilbert space of entire functions and one example of a continuous representation. Attention is devoted to the purely functional characterization of the continuous representation by means of the reproducing kernel as a special case of Aronszajn's general theory. Properties of various operators in a continuous representation are carefully defined.