论文标题
无限制的自我关节操作员的散射矩阵的近距离奇异自我扰动
Scattering Matrices for Close Singular Selfadjoint Perturbations of Unbounded Selfadjoint Operators
论文作者
论文摘要
在本文中，我们考虑了一个无界的自我关节运算符$ a $及其在同一希尔伯特空间中的自adexhixhexhixhexhighthexhighthexhightercations $ \ Mathcal {h} $。正如S.Albeverio和P. Kurosov（2000）一样，我们称为自addodchoint Operator $ a_ {1} $，如果$ a_ {1} $和{a}的单数扰动具有不同的域$ \ Mathcal {d}（d}（d}（d}（d}（a））， $ \ MATHCAL {D}（A）\ CAP \ MATHCAL {D}（A_ {1}）$。假设$ a $具有绝对连续的频谱，并且分解$ r_ {z {z}（a_ {1}）-r_ {z}（z}（a）的$ a_ {1} $和非现实$ z $的$ a $ a $ a $ a $ z $是痕迹类运算符，是一个痕迹的运算符，我们找到了零件的明确表达式，我们为$ $ a y} $ ________________该对的分解的公式。作为例证，我们在$ l_ {2} \ left（\ Mathbf {r} _ {3} \ right）$中找到标准定义的拉普拉斯操作员的散射矩阵及其以无限范围潜能的形式的奇异扰动。
In this paper, we consider an unbounded selfadjoint operator $A$ and its selfadjoint perturbations in the same Hilbert space $\mathcal{H}$. As S.Albeverio and P. Kurosov (2000), we call a selfadjoint operator $A_{1}$ the singular perturbation of $A$ if $A_{1}$ and {A} have different domains $\mathcal{D}(A),\mathcal{D}(A_{1})$ but $A=A_{1}$ on $\mathcal{D}(A)\cap\mathcal{D}(A_{1})$. Assuming that $A$ has absolutely continuous spectrum and the difference of resolvents $R_{z}(A_{1}) -R_{z}(A)$ of $A_{1}$ and $A$ for non-real $z$ is a trace class operator we find the explicit expression for the scattering matrix for the pair $A, A_{1}$ through the constituent elements of the Krein formula for the resolvents of this pair. As an illustration, we find the scattering matrix for the standardly defined Laplace operator in $L_{2}\left(\mathbf{R}_{3}\right)$ and its singular perturbation in the form of an infinite sum of zero-range potentials.
