Please use this identifier to cite or link to this item: http://dspace.onu.edu.ua:8080/handle/123456789/11062
Title: Dirac–Krein Systems on Star Graphs
Authors: Adamyan, Vadym M.
Langer, H.
Tretter, C.
Winklmeier, M.
Адамян, Вадим Мовсесович
Citation: Integral Equations and Operator Theory
Issue Date: 26-Aug-2016
Keywords: Dirac operator
Dirac–Krein system
star graph
Krein’s resolvent
formula
trace formula
dislocation index
Series/Report no.: ;Vol. 86, Issue 1
Abstract: We study the spectrum of a self-adjoint Dirac–Krein operator with potential on a compact star graph 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 G. Special attention is paid to Robin matching conditions with parameter τ ∈ R∪{∞}. 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 τ , and interlacing properties of the eigenvalues, and prove a trace formula. Moreover, we show that, asymptotically for R → ∞, the difference of the number of eigenvalues in the intervals [0,R) and [−R, 0) deviates from some integer κ0, which we call dislocation index, at most by n+2.
URI: http://dspace.onu.edu.ua:8080/handle/123456789/11062
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