{"f47f251990b9280b3f0ddddf1ed72328elkepeter":{"DOI":"","ISBN":"","ISSN":"","URL":"","abstract":"\"The authors study eigenvalues of the Laplacian (the negative second derivative operator) of a compact metric graph equipped with complex δ-vertex conditions. More precisely:\r\n(a) a continuity condition is imposed at each vertex;\r\n(b) on a selected set of vertices R, the vertex conditions\r\n∑e∼vj∂∂νf|e(vj)+αjf(vj)=0\r\nare imposed, where the interaction strengths αj are complex parameters and the derivative is taken in the direction towards the vertex; and\r\n(c) on the vertices in ∖R, Kirchhoff vertex conditions are imposed.\r\n\r\n   The main focus in this article is on the asymptotic behavior of the (purely discrete) spectrum as certain coefficients αj tend to ∞ in the complex plane. If m of these coefficients tend to infinity within some sector in the open left half-plane while the remaining coefficients tend to infinity in a way such that Reαj remains bounded from below, then the authors show that exactly m eigenvalues diverge away from the positive real semi-axis. Moreover, they provide the asymptotics of these eigenvalues; the leading term is quadratic in the αj, with a coefficient depending on the vertex degrees. \r\n   As variational principles are not available for the study of such non-self-adjoint problems, the authors use a Birman-Schwinger type characterization of eigenvalues in terms of a parameter-dependent Dirichlet-to-Neumann matrix (or Titchmarsh-Weyl function). In addition, they also obtain estimates for the numerical range and the eigenvalues.\"","annote":"","author":[{"family":"Kennedy","given":"James B."},{"family":"Lang","given":"Robin"}],"citation-label":"kennedy2020eigenvalues","collection-editor":[],"collection-title":"","container-author":[],"container-title":"Portugaliae Mathematica. A Journal of the Portuguese Mathematical Society","documents":[],"edition":"","editor":[],"event-date":{"date-parts":[["2020"]],"literal":"2020"},"event-place":"","id":"f47f251990b9280b3f0ddddf1ed72328elkepeter","interhash":"002810b89f26ce8985d6810f83206ff3","intrahash":"f47f251990b9280b3f0ddddf1ed72328","issue":"2","issued":{"date-parts":[["2020"]],"literal":"2020"},"keyword":"iadm Kennedy Laplacians Lang quantum graph","misc":{"language":"English"},"note":"","number":"2","number-of-pages":"28","page":"133-161","page-first":"133","publisher":"","publisher-place":"","status":"","title":"On the eigenvalues of quantum graph Laplacians with large complex δ couplings.","type":"article-journal","username":"elkepeter","version":"","volume":"77"},"4ffea2310033ddac3ec52d77b210ba8aelkepeter":{"DOI":"10.18419/opus-11428","ISBN":"","ISSN":"","URL":"","abstract":"","annote":"","author":[{"family":"Lang","given":"Robin"}],"citation-label":"lang2021eigenvalues","collection-editor":[],"collection-title":"","container-author":[],"container-title":"","documents":[],"edition":"","editor":[],"event-date":{"date-parts":[["2021"]],"literal":"2021"},"event-place":"Stuttgart","genre":"PhD dissertation","id":"4ffea2310033ddac3ec52d77b210ba8aelkepeter","interhash":"1d9d3866bfc31d5f2d0217f6177e7cc4","intrahash":"4ffea2310033ddac3ec52d77b210ba8a","issue":"","issued":{"date-parts":[["2021"]],"literal":"2021"},"keyword":"iadm graphs Lang quantum Robin Laplacian","misc":{"supervisorgnd":"1095129147","language":"English","eventdate":"2020-12-01","supervisor":"Weidl, Timo","doi":"10.18419/opus-11428"},"note":"","number":"","page":"","page-first":"","publisher":"Universität Stuttgart","publisher-place":"Stuttgart","status":"","title":"On the eigenvalues of the non-self-adjoint Robin Laplacian on bounded domains and compact quantum graphs.","type":"thesis","username":"elkepeter","version":"","volume":""}}