谱位移函数的Weak Convergence的一种抽象方法及其在多维薛定谔算子中的应用
An Abstract Approach to Weak Convergence of Spectral Shift Functions and Applications to Multi-Dimensional Schrödinger Operators
摘要：
我们研究了希尔伯特空间中与自伴算子$(H_j, H_{0,j})$的抽象对相关联的谱移函数序列$ξ(\cdot;H_j,H_{0,j})$收敛到与极限希尔伯特空间$\cH_j$中的一对$(H,H_0)$相关联的极限谱移函数$ξ(\cdot;H,H_0)$作为$j\to\infty$的方式（在多维薛定谔算子的具体应用中模仿无限体积极限）。我们的技术依赖于Fredholm行列式的方法结合一定的测量理论的事实。特别是，我们证明了文献中谱移函数的先验模糊收敛结果实际上扩展到弱收敛的概念。更确切地说，在多维薛定谔算子的具体情况下，在一系列区域$Ω_j$耗尽$\bbR^n$作为$j\to\infty$，我们扩展了相关的谱移函数的收敛性从模糊到弱收敛，也从Dirichlet边界条件更一般的自伴边界条件$\partialΩ_j$。
英文摘要：
We study the manner in which a sequence of spectral shift functions $ξ(\cdot;H_j,H_{0,j})$ associated with abstract pairs of self-adjoint operators $(H_j, H_{0,j})$ in Hilbert spaces $\cH_j$, $j\in\bbN$, converge to a limiting spectral shift function $ξ(\cdot;H,H_0)$ associated with a pair $(H,H_0)$ in the limiting Hilbert space $\cH$ as $j\to\infty$ (mimicking the infinite volume limit in concrete applications to multi-dimensional Schrödinger operators). Our techniques rely on a Fredholm determinant approach combined with certain measure theoretic facts. In particular, we show that prior vague convergence results for spectral shift functions in the literature actually extend to the notion of weak convergence. More precisely, in the concrete case of multi-dimensional Schrödinger operators on a sequence of domains $Ω_j$ exhausting $\bbR^n$ as $j\to\infty$, we extend the convergence of associated spectral shift functions from vague to weak convergence and also from Dirichlet boundary conditions to more general self-adjoint boundary conditions on $\partialΩ_j$.
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