On the eigenvalues of quantum graph Laplacians with large complex δ couplings.
J. Kennedy, and R. Lang. Portugaliae Mathematica. A Journal of the Portuguese Mathematical Society, 77 (2):
133-161(2020)
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."
%0 Journal Article
%1 kennedy2020eigenvalues
%A Kennedy, James B.
%A Lang, Robin
%D 2020
%J Portugaliae Mathematica. A Journal of the Portuguese Mathematical Society
%K Kennedy iadm from:elkepeter Laplacians Lang quantum graph
%N 2
%P 133-161
%T On the eigenvalues of quantum graph Laplacians with large complex δ couplings.
%V 77
%X "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."
@article{kennedy2020eigenvalues,
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."},
added-at = {2021-10-13T13:54:00.000+0200},
author = {Kennedy, James B. and Lang, Robin},
biburl = {https://puma.ub.uni-stuttgart.de/bibtex/2f47f251990b9280b3f0ddddf1ed72328/mathematik},
interhash = {002810b89f26ce8985d6810f83206ff3},
intrahash = {f47f251990b9280b3f0ddddf1ed72328},
journal = {Portugaliae Mathematica. A Journal of the Portuguese Mathematical Society},
keywords = {Kennedy iadm from:elkepeter Laplacians Lang quantum graph},
language = {English},
number = 2,
pages = {133-161},
timestamp = {2021-10-13T11:54:00.000+0200},
title = {On the eigenvalues of quantum graph Laplacians with large complex δ couplings.},
volume = 77,
year = 2020
}