Dirac–Krein Systems on Star Graphs
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- 作者：V. Adamyan ; H. Langer ; C. Tretter ; M. Winklmeier
- 关键词：Dirac operator ; Dirac–Krein system ; star graph ; Krein’s resolvent formula ; trace formula ; dislocation index
- 刊名：Integral Equations and Operator Theory
- 出版年：2016
- 出版时间：September 2016
- 年：2016
- 卷：86
- 期：1
- 页码：121-150
- 全文大小：777 KB
- 刊物类别：Mathematics and Statistics
- 刊物主题：Mathematics
Analysis - 出版者：Birkh盲user Basel
- ISSN：1420-8989
- 卷排序：86
文摘
We study the spectrum of a self-adjoint Dirac–Krein operator with potential on a compact star graph \({\mathcal{G}}\) with a finite number n of edges. This operator is defined by a Dirac–Krein differential expression with summable matrix potentials on each edge, by self-adjoint boundary conditions at the outer vertices, and by a self-adjoint matching condition at the common central vertex of \({\mathcal{G}}\). Special attention is paid to Robin matching conditions with parameter \({\tau \in \mathbb{R} \cup\{\infty\}}\). Choosing the decoupled operator with Dirichlet condition at the central vertex as a reference operator, we derive Krein’s resolvent formula, introduce corresponding Weyl–Titchmarsh functions, study the multiplicities, dependence on \({\tau}\), and interlacing properties of the eigenvalues, and prove a trace formula. Moreover, we show that, asymptotically for \({R\to \infty}\), the difference of the number of eigenvalues in the intervals \({[0,R)}\) and \({[-R,0)}\) deviates from some integer \({\kappa_0}\), which we call dislocation index, at most by \({n+2}\).