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:
(a) a continuity condition is imposed at each vertex;
(b) on a selected set of vertices R, the vertex conditions
∑e∼vj∂∂νf|e(vj)+αjf(vj)=0
are imposed, where the interaction strengths αj are complex parameters and the derivative is taken in the direction towards the vertex; and
(c) on the vertices in ∖R, Kirchhoff vertex conditions are imposed.
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.
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."
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