Dirac-Krein systems on star graphs

Adamyan, V.; Langer, Heinz; Tretter, Christiane; Winklmeier, Monika (2016). Dirac-Krein systems on star graphs. Integral equations and operator theory, 86(1), pp. 121-150. Birkhäuser 10.1007/s00020-016-2311-4

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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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