Adamyan, V.; Langer, Heinz; Tretter, Christiane; Winklmeier, Monika (2016). DiracKrein systems on star graphs. Integral equations and operator theory, 86(1), pp. 121150. Birkhäuser 10.1007/s0002001623114

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We study the spectrum of a selfadjoint DiracKrein operator with potential on a compact star graph G with a finite number n of edges. This operator is defined by a DiracKrein differential expression with summable matrix potentials on each edge, by selfadjoint boundary conditions at the outer vertices, and by a selfadjoint 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 WeylTitchmarsh 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.
Item Type: 
Journal Article (Original Article) 

Division/Institute: 
08 Faculty of Science > Department of Mathematics and Statistics > Institute of Mathematics 
UniBE Contributor: 
Langer, Heinz; Tretter, Christiane and Winklmeier, Monika 
Subjects: 
500 Science > 510 Mathematics 
ISSN: 
0378620X 
Publisher: 
Birkhäuser 
Language: 
English 
Submitter: 
Olivier Bernard Mila 
Date Deposited: 
09 Aug 2017 13:06 
Last Modified: 
01 Oct 2017 02:30 
Publisher DOI: 
10.1007/s0002001623114 
ArXiv ID: 
1608.05865v1 
BORIS DOI: 
10.7892/boris.99765 
URI: 
https://boris.unibe.ch/id/eprint/99765 