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Anomalous nodal count and singularities in the dispersion relation of honeycomb graphs. (English) Zbl 1335.81078

The authors consider a Schrödinger operator on a metric graph with a piecewise continuous potential subject to \(\delta\)-type matching conditions at the vertices. The nodal surplus of a generic eigenfunction corresponding to the \(n\)-th eigenvalue \(\lambda_n\) is defined as \(\sigma_n := \phi_n - (n - 1)\), where \(\phi_n\) is the number of internal zeros of the eigenfunction. By a known result the relation \(0 \leq \sigma_n \leq \beta\) holds for all \(n\), where \(\beta\) is the first Betti number of the graph. The authors consider a special class of graphs, so called bi-dendral graphs, which arise when two copies of a tree graph are glued together by identifying corresponding leaves and imposing Neumann conditions at the new, intermediate vertices of degree two. They show that for each bi-dendral graph the nodal surplus satisfies \(1 \leq \sigma_n \leq \beta - 1\) at each generic eigenpair. The same result holds when so-called anti-Neumann conditions are imposed at certain of the intermediate vertices. In addition the authors study internal critical points of the eigenvalues \(\lambda_n (\alpha)\) (as functions of \(\alpha\)) of the corresponding magnetic Schrödinger operator with magnetic flux \(\alpha\).

MSC:

81Q35 Quantum mechanics on special spaces: manifolds, fractals, graphs, lattices
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
81U30 Dispersion theory, dispersion relations arising in quantum theory
47A48 Operator colligations (= nodes), vessels, linear systems, characteristic functions, realizations, etc.
78A25 Electromagnetic theory (general)
46F10 Operations with distributions and generalized functions
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