论文标题
Sturm-Liouville操作员的奇异边界条件通过扰动理论
Singular Boundary Conditions for Sturm--Liouville Operators via Perturbation Theory
论文作者
论文摘要
我们表明，可以通过添加性奇异形式有界的自我吸引的等级扰动，例如等于缺陷指数（例如$ d \ in \ in \ {1,2 \} $），可以通过添加性奇异形式有界的自我扰动来获得具有通用极限端点的半结合sturm- liouville操作员的所有自动参与扩展。这种表征概括了具有常规端点的半结合的Sturm- liouville操作员的众所周知的类似物。明确地，最小运算符的每个自我伴侣扩展可以写成\ begin {align*} \ boldsymbol {a}_θ= \ boldsymbol {a} _0+{\ bf bf b}θθ{\ bf b}^*，\ end end {align*}其中$ \ boldsymbol {a} _0 $是一个杰出的自我接触延伸，$ $ $ $ $ $ usex lination a in a lination a lination a lination in lination n in a lination lination in lination in a in a in niment lination in $ \ mathbb {c}^d $。从某种意义上说，它不属于基础的希尔伯特空间，而是相对于$ \ boldsymbol {a} _0 $，即属于$ \ Mathcal {h} _ {-1}（\ boldsymbol {a a} _0）$。对称操作员的边界三重和兼容边界对的构建可确保扰动明确定义，并且自相关扩展在与自相关关系$θ$的一对一对应关系中。 例如，获得了经典对称的雅各比微分方程（具有两个极限端点）的自相关扩展，并用边界三元组和扰动理论的工具分析了它们的光谱。
We show that all self-adjoint extensions of semi-bounded Sturm--Liouville operators with general limit-circle endpoint(s) can be obtained via an additive singular form bounded self-adjoint perturbation of rank equal to the deficiency indices, say $d\in\{1,2\}$. This characterization generalizes the well-known analog for semi-bounded Sturm--Liouville operators with regular endpoints. Explicitly, every self-adjoint extension of the minimal operator can be written as \begin{align*} \boldsymbol{A}_Θ=\boldsymbol{A}_0+{\bf B}Θ{\bf B}^*, \end{align*} where $\boldsymbol{A}_0$ is a distinguished self-adjoint extension and $Θ$ is a self-adjoint linear relation in $\mathbb{C}^d$. The perturbation is singular in the sense that it does not belong to the underlying Hilbert space but is form bounded with respect to $\boldsymbol{A}_0$, i.e. it belongs to $\mathcal{H}_{-1}(\boldsymbol{A}_0)$. The construction of a boundary triple and compatible boundary pair for the symmetric operator ensure that the perturbation is well-defined and self-adjoint extensions are in a one-to-one correspondence with self-adjoint relations $Θ$. As an example, self-adjoint extensions of the classical symmetric Jacobi differential equation (which has two limit-circle endpoints) are obtained and their spectra are analyzed with tools both from the theory of boundary triples and perturbation theory.
