JournalspmVol. 77, No. 2pp. 133–161

On the eigenvalues of quantum graph Laplacians with large complex δ\delta couplings

  • James B. Kennedy

    Universidade de Lisboa, Portugal
  • Robin Lang

    Universität Stuttgart, Germany
On the eigenvalues of quantum graph Laplacians with large complex $\delta$ couplings cover
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Abstract

We study the location of the spectrum of the Laplacian on compact metric graphs with complex Robin-type vertex conditions, also known as δ\delta conditions, on some or all of the graph vertices. We classify the eigenvalue asymptotics as the complex Robin parameter(s) diverge to \infty in C\mathbb C: for each vertex vv with a Robin parameter αC\alpha \in \mathbb C for which Reα\alpha \to -\infty sufficiently quickly, there exists exactly one divergent eigenvalue, which behaves like α2/degv2-\alpha^2/\mathrm {deg} v^2, while all other eigenvalues stay near the spectrum of the Laplacian with a Dirichlet condition at vv; if Reα\mathrm {Re} \alpha remains bounded from below, then all eigenvalues stay near the Dirichlet spectrum. Our proof is based on an analysis of the corresponding Dirichlet-to-Neumann matrices (Titchmarsh–Weyl MM-functions). We also use sharp trace-type inequalities to prove estimates on the numerical range and hence on the spectrum of the operator, which allow us to control both the real and imaginary parts of the eigenvalues in terms of the real and imaginary parts of the Robin parameter(s).

Cite this article

James B. Kennedy, Robin Lang, On the eigenvalues of quantum graph Laplacians with large complex δ\delta couplings. Port. Math. 77 (2020), no. 2, pp. 133–161

DOI 10.4171/PM/2047