约斯特与派斯主题变奏曲
Variations on a Theme of Jost and Pais
摘要：
我们探讨了在何种程度上的一个著名的公式，由于约斯特和派斯，减少了Fredholm扰动行列式与薛定谔算子的半线上的一个简单的Wronski行列式的适当的分配解决方案的基本薛定谔方程，推广到更高的维度。在这个多维扩展中，半线被开集$Ω\subset\bbR^n$，$n\in\bbN$，$n\geq 2$所取代，其中$Ω$具有满足一定正则性条件的紧致非空边界$\partialΩ$。我们的变体涉及的扰动行列式的比例对应于Dirichlet和诺伊曼边界条件$\partialΩ$，并调用相应的Dirichlet-诺伊曼映射。其结果是，我们成功地减少了一定比例的修改Fredholm扰动行列式与运营商在$L^2(Ω; d^n x)$，$n\in\bbN$，修改弗雷德霍姆行列式与运营商在$L^2(\partialΩ; d^{n-1}σ)$，$n\geq 2$。Birman-Schwinger原理和本征值计数函数的应用进行了讨论。
英文摘要：
We explore the extent to which a variant of a celebrated formula due to Jost and Pais, which reduces the Fredholm perturbation determinant associated with the Schrödinger operator on a half-line to a simple Wronski determinant of appropriate distributional solutions of the underlying Schrödinger equation, generalizes to higher dimensions. In this multi-dimensional extension the half-line is replaced by an open set $Ω\subset\bbR^n$, $n\in\bbN$, $n\geq 2$, where $Ω$ has a compact, nonempty boundary $\partialΩ$ satisfying certain regularity conditions. Our variant involves ratios of perturbation determinants corresponding to Dirichlet and Neumann boundary conditions on $\partialΩ$ and invokes the corresponding Dirichlet-to-Neumann map. As a result, we succeed in reducing a certain ratio of modified Fredholm perturbation determinants associated with operators in $L^2(Ω; d^n x)$, $n\in\bbN$, to modified Fredholm determinants associated with operators in $L^2(\partialΩ; d^{n-1}σ)$, $n\geq 2$. Applications involving the Birman-Schwinger principle and eigenvalue counting functions are discussed.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
