Boundary Data Maps and Krein's Resolvent Formula for Sturm-Liouville Operators on a Finite Interval
Boundary Data Maps and Krein's Resolvent Formula for Sturm-Liouville Operators on a Finite Interval
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有限区间上 Sturm-Liouville 算子的边界数据图和 Krein 求解公式
DOI：
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发表时间：
2012
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通讯作者：
M. Zinchenko
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作者：
Stephen Clark;F. Gesztesy;Roger Nichols;M. Zinchenko
我们继续研究的边界数据映射，即广义谱参数依赖Dirichlet-诺依曼映射（三系数）Sturm-Liouville算子的有限区间$（a，B），更一般的边界条件。虽然边界数据图的早期研究集中在一般分离边界条件的情况下，在$a$和$B$，目前的工作发展了一个统一的处理所有可能的自伴边界条件（即，分离的和不分离的）。
在本文中，我们描述了基本的极小Sturm-Liouville算子的自伴扩张与Krein预解式的关系（在边界条件方面参数化），在一定程度上强调Krein扩张，发展自伴扩张的预解式差的基本迹公式，特别是，在相关的谱移函数方面，并描述了所有自伴扩张的各种参数化之间的联系，包括与冯诺依曼的基本参数化之间的精确关系，在亏量子空间之间的酉映射。
We continue the study of boundary data maps, that is, generalizations of spectral parameter dependent Dirichlet-to-Neumann maps for (three-coefficient) Sturm-Liouville operators on the finite interval $(a,b)$, to more general boundary conditions. While earlier studies of boundary data maps focused on the case of general separated boundary conditions at $a$ and $b$, the present work develops a unified treatment for all possible self-adjoint boundary conditions (i.e., separated as well as non-separated ones).
In the course of this paper we describe the connections with Krein's resolvent formula for self-adjoint extensions of the underlying minimal Sturm-Liouville operator (parametrized in terms of boundary conditions), with some emphasis on the Krein extension, develop the basic trace formulas for resolvent differences of self-adjoint extensions, especially, in terms of the associated spectral shift functions, and describe the connections between various parametrizations of all self-adjoint extensions, including the precise relation to von Neumann's basic parametrization in terms of unitary maps between deficiency subspaces.