Variations on a theme of Jost and Pais

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• 作者：Fritz Gesztesy ; Marius Mitrea ; Maxim Zinchenko
• 关键词：Fredholm determinants ; Non-self-adjoint operators ; Multi-dimensional Schrö ; dinger operators ; Dirichlet-to-Neumann maps
• 刊名：Journal of Functional Analysis
• 出版年：2007
• 出版时间：15 December 2007
• 年：2007
• 卷：253
• 期：2
• 页码：399-448
• 全文大小：411 K

文摘

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 , , n2, where Ω has a compact, nonempty boundary ∂Ω satisfying certain regularity conditions. Our variant involves ratios of perturbation determinants corresponding to Dirichlet and Neumann boundary conditions on ∂Ω 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 L²(Ω;dⁿx), , to modified Fredholm determinants associated with operators in L²(∂Ω;dⁿ⁻¹σ), n2. Applications involving the Birman–Schwinger principle and eigenvalue counting functions are discussed.