Dirac–Krein Systems on Star Graphs
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2016-08-26
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Анотація
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.
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Ключові слова
Dirac operator, Dirac–Krein system, star graph, Krein’s resolvent, formula, trace formula, dislocation index
Бібліографічний опис
Integral Equations and Operator Theory