标题： 关于平方可积解的个数与对称一阶微分方程组的自伴性 显示英文标题

标题： On the number of square integrable solutions and self-adjointness of symmetric first order systems of differential equations

摘要： 本文的主要目的是研究区间 $I$上对称一阶系统 $$ Jf'+Bf=\lambda {\cal H} f $$的形式缺陷指标 ${\cal N}_{\pm}(I)$，其中 $I=\mathbb{R}$ 或 $I=\mathbb{R}_\pm.$ 此处 $J,B,{cal H}$是 $n\times n$ 矩阵值函数，而哈密顿量 ${\cal H}\ge 0$ 即使在处处也可能奇异。 我们得到两个结果，使得该系统具有最小数${\cal N}_\pm(\mathbb{R})=0$（分别为${\cal N}_\pm(\mathbb{R}_\pm)=n$）以及它们的最大性准则${\cal N}_{\pm}(\mathbb{R}_+)=2n.$。也提出了关于规范系统具有中间数${\cal N}_\pm(\mathbb{R}_+)$的一些条件。 我们还得到了对二阶Sturm-Liouville型方程的著名Titchmarsh-Sears定理的推广。 这些结果包括Lidskii和Krein的结果作为特殊情况。 需要注意的是，一般来说，上述系统并不产生一个算子，而是产生希尔伯特空间中的一个对称线性关系。 这些关系被详细研究。 作为副产品，我们得到了Kogan和Rofe-Beketov（Proc. Roy. Soc. Edinb. 74 (1974/75)）论文的主要结果的非常简短的证明（以及其推广），以及规范系统的准正则性的准则。 这涵盖了Kac-Krein定理以及Kogan和Rofe-Beketov所引用论文中的一些结果。 显示英文摘要

摘要： The main purpose of this paper is to investigate the formal deficiency indices ${\cal N}_{\pm}(I)$ of a symmetric first order system $$ Jf'+Bf=\lambda {\cal H} f $$ on an interval $I$, where $I=\mathbb{R}$ or $I=\mathbb{R}_\pm.$ Here $J,B,{cal H}$ are $n\times n$ matrix valued functions and the Hamiltonian ${\cal H}\ge 0$ may be singular even everywhere. We obtain two results for such a system to have minimal numbers ${\cal N}_\pm(\mathbb{R})=0$ (resp. ${\cal N}_\pm(\mathbb{R}_\pm)=n$) and a criterion for their maximality ${\cal N}_{\pm}(\mathbb{R}_+)=2n.$ Some conditions for a canonical system to have intermediate numbers ${\cal N}_\pm(\mathbb{R}_+)$ are presented, too. We also obtain a generalization of the well-known Titchmarsh-Sears theorem for second order Sturm-Liouville type equations. This contains results due to Lidskii and Krein as special cases. It is important to note that in general the above system does not give rise to an operator but rather to a symmetric linear relation in a Hilbert space. These relations are investigated in detail. As a byproduct we obtain very short proofs of (generalizations of) the main results of a paper by Kogan and Rofe-Beketov (Proc. Roy. Soc. Edinb. 74 (1974/75))as well as a criterion for the quasi-regularity of canonical systems. This covers the Kac-Krein theorem and some results from the quoted paper of Kogan and Rofe-Beketov.