论文标题
在一类积分系统上
On a class of integral systems
论文作者
论文摘要
我们研究具有两个给定的非缩减功能的二维积分系统的光谱问题$ r_1 $，$ r_2 $在间隔$ [0，b）$上，这是kerin string的概括。与此系统关联的是最大线性关系$ t _ {\ max} $和最小线性关系$ t _ {\ min} $中的空间$ l^2（r_2）$，这些$由$ t _ {\ max} = t _ {\ max} = t _ {\ min min}^*$连接。结果表明，该系统的$ b $处的极限点条件相当于线性关系的强限点条件$ t _ {\ max} $。在限制圈情况下，强极点条件无法在$ t _ {\ max} $上固定，但是在子空间$ t_n^*$ of $ t _ {\ max} $上以$ b $为特征的$ t _ {\ max} $。该积分系统的主要titchmarsh-weyl系数的概念均在限点情况下和极限圆圈中引入。构建了限制点情况下的线性关系$ t _ {\ max} $（在极限圆圈中为$ t_ {n}^*$）的边界三元三元构建，并显示相应的Weyl函数与集成系统的主titchmarsh-weyl系数重合。双重积分系统的概念是通过逆转$ R_1 $和$ R_2 $的订单来介绍的。结果表明，直接的主要titchmarsh -weyl系数$ q $ $ q $和$ \ widehat q $，双积分系统与平等$λ\ widehat q（λ）= -1/q（λ）$相关。
We study spectral problems for two--dimensional integral system with two given non-decreasing functions $R_1$, $R_2$ on an interval $[0,b)$ which is a generalization of the Krein string. Associated to this system are the maximal linear relation $T_{\max}$ and the minimal linear relation $T_{\min}$ in the space $L^2(R_2)$ which are connected by $T_{\max}=T_{\min}^*$. It is shown that the limit point condition at $b$ for this system is equivalent to the strong limit point condition for the linear relation $T_{\max}$. In the limit circle case the strong limit point condition fails to hold on $T_{\max}$ but it is still satisfied on a subspace $T_N^*$ of $T_{\max}$ characterized by the Neumann boundary condition at $b$. The notion of the principal Titchmarsh-Weyl coefficient of this integral system is introduced both in the limit point case and in the limit circle case. Boundary triples for the linear relation $T_{\max}$ in the limit point case (and for $T_{N}^*$ in the limit circle case) are constructed and it is shown that the corresponding Weyl function coincides with the principal Titchmarsh-Weyl coefficient of the integral system. The notion of the dual integral system is introduced by reversing the order of $R_1$ and $R_2$. It is shown that the principal Titchmarsh-Weyl coefficients $q$ and $\widehat q$ of the direct and the dual integral systems are related by the equality $λ\widehat q(λ) = -1/q(λ)$ both in the regular and the singular case.
