Singular quadratic functionals
Singular quadratic functionals
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DOI：
10.1090/s0002-9947-1936-1501873-7
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发表时间：
1936-02
影响因子：
1.3
通讯作者：
M. Morse;Walter Leighton
中科院分区：
文献类型：
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作者：
M. Morse;Walter Leighton
导论.肯布尔[1]结合量子力学对（1.1）型奇异二次函数作了简要的研究。肯布尔的结果具有相对有限的性质。另一方面，哈代，Littlewood，和Pólya，f在第七章[1 ]中研究了特殊的例子，这些泛函采用特殊的方法。本文的发展显然对（1.1）型奇异二次泛函极小化问题给出了一个系统的方法。特别地，所得结果包含和推广了H.L.P.的定理（254）和（253），见例9.2和12.1。作者承认有各种类型的比较曲线。结果显示在惊人的方式如何存在的最小值取决于类的曲线承认。这个问题需要一个重塑的共枕点理论和引进一个新的条件称为奇异性条件。勒贝格积分或其扩展使用整个。本文的结果将用于推广特征根理论及相关边值问题的求解。
Introduction. Singular quadratic functions of the type (1.1) have been briefly investigated by Kemble [l ] in connection with Quantum Mechanics. The results of Kemble are of a relatively restricted character. On the other hand, Hardy, Littlewood, and Pólya,f in Chapter VII of [1 ] have studied special examples of these functionals employing special methods. The developments of this paper apparently give a systematic approach to the problem of minimizing singular quadratic functionals of the type (1.1). In particular, the results obtained include and generalize Theorems (254) and (253) of H.L.P. See Examples 9.2 and 12.1 of this paper. The authors have admitted various classes of comparison curves. The results show in striking fashion how the existence of the minimum depends upon the classes of curves admitted. The problem requires a remodeling of the conjugate point theory and an introduction of a new condition called the singularity condition. Lebesgue integrals or their extensions are used throughout. The results of this paper will be applied to extend the theory of characteristic roots and solutions of the related boundary problems.