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Differences between Robin and Neumann eigenvalues on metric graphs

Ram Band, Holger Schanz, Gilad Sofer

Abstract

We consider the Laplacian on a metric graph, equipped with Robin ($δ$-type) vertex condition at some of the graph vertices and Neumann-Kirchhoff condition at all others. The corresponding eigenvalues are called Robin eigenvalues, whereas they are called Neumann eigenvalues if the Neumann-Kirchhoff condition is imposed at all vertices. The sequence of differences between these pairs of eigenvalues is called the Robin-Neumann gap. We prove that the limiting mean value of this sequence exists and equals a geometric quantity, analogous to the one obtained for planar domains. Moreover, we show that the sequence is uniformly bounded and provide explicit upper and lower bounds. We also study the possible accumulation points of the sequence and relate those to the associated probability distribution of the gaps. To prove our main results, we prove a local Weyl law, as well as explicit expressions for the second moments of the eigenfunction scattering amplitudes.

Differences between Robin and Neumann eigenvalues on metric graphs

Abstract

We consider the Laplacian on a metric graph, equipped with Robin (-type) vertex condition at some of the graph vertices and Neumann-Kirchhoff condition at all others. The corresponding eigenvalues are called Robin eigenvalues, whereas they are called Neumann eigenvalues if the Neumann-Kirchhoff condition is imposed at all vertices. The sequence of differences between these pairs of eigenvalues is called the Robin-Neumann gap. We prove that the limiting mean value of this sequence exists and equals a geometric quantity, analogous to the one obtained for planar domains. Moreover, we show that the sequence is uniformly bounded and provide explicit upper and lower bounds. We also study the possible accumulation points of the sequence and relate those to the associated probability distribution of the gaps. To prove our main results, we prove a local Weyl law, as well as explicit expressions for the second moments of the eigenfunction scattering amplitudes.
Paper Structure (21 sections, 18 theorems, 108 equations, 12 figures)

This paper contains 21 sections, 18 theorems, 108 equations, 12 figures.

Table of Contents

  1. Introduction
  2. Basic definitions and notations
  3. Main results
  4. Review of tools and methods for proofs
  5. Expressing the Robin-Neumann gap via eigenfunction values
  6. Scattering formalism and the secular equation
  7. The secular manifold
  8. Integrating over the secular manifold -- the Barra-Gaspard measure
  9. Proof of Theorem \ref{['thm:Lipschitz_RNG']}
  10. Proof of Theorem \ref{['thm:Weyl-law']}
  11. Proof of Theorem \ref{['thm:mean_RNG']}
  12. Proof and discussion of Theorems \ref{['thm:probability']} and \ref{['thm:converging_subsequence']}
  13. Proof and Discussion of Theorem \ref{['thm:explicit_bounds']}
  14. Proof of Theorem \ref{['thm:explicit_bounds']}
  15. Discussion of Lemma \ref{['lemma:explicit_bounds']} and Theorem \ref{['thm:explicit_bounds']}
  16. ...and 6 more sections

Key Result

Theorem 1.3

The Cesàro mean of the Robin-Neumann gap exists and satisfies where $\left|\Gamma\right|$ is the total length of the graph and $\deg\left(v\right)$ is the degree of the vertex $v$.

Figures (12)

  • Figure 1.1: The Robin eigenvalues $\lambda_{n}\left(\sigma\right)$ for a star graph, along with the Robin-Neumann gap $d_{5}\left(4.5\right)$. The Robin vertices are marked in red.
  • Figure 1.2: Scatter plot of the first $2,500$ Robin-Neumann gaps for a star graph with four edges and Robin condition at the central vertex, normalized so that $\left\langle d\right\rangle _{n}\left(\sigma\right)=1$. The light blue line is a running average and the blue lines on top of it are the analytic results from Equations (\ref{['eq:RNG_mean_const']}), (\ref{['eq:RNG_mean_arctan']}). The red dashed line is the first upper bound presented in Equation (\ref{['eq:explicit_bounds']}), while the solid red line is the finer upper bound appearing in Equation (\ref{['eq:upperbound']}) (under the star decomposition described in Subsection \ref{['subsec:Explicit-estimate-of']}).
  • Figure 1.3: Star decomposition of a tetrahedron graph. The vertices and edges of the original graph are shown in black. The small blue dots correspond to the auxiliary vertices. The star graph around vertex 4 is highlighted in red.
  • Figure 6.1: The torus flow $\phi\left(k\right)=\left(2k,k\right)$ on $\mathbb{T}^{2}$ as an example of rationally dependent entries (here $D=1$). After the change of coordinates $\left(\varphi_{1},\varphi_{2}\right)=\frac{1}{3}\left(\kappa_{1}+\kappa_{2},\kappa_{1}-2\kappa_{2}\right)$, we get the flow $\tilde{\phi}\left(k\right)=\left(k,0\right)$, which is dense in the first component and constant in the second component.
  • Figure 6.2: RNG (black points) for an equilateral star graph with Robin condition at the central vertex, scaled so that $\langle d\rangle_{n}=1$ .The fluctuating light blue line is a running average and the blue lines on top of it are the analytic results from Equations (\ref{['eq:RNG_mean_const']}), (\ref{['eq:RNG_mean_arctan']}). The red dashed and full lines are the analytic bounds of Equation (\ref{['eq:explicit_bounds']}), (\ref{['eq:upperbound']}). Since many of the states vanish at the central vertex, their corresponding RNG is zero. The RNG accumulates at two particular values, and does not get close to the mean value.
  • ...and 7 more figures

Theorems & Definitions (46)

  • Remark 1.1
  • Definition 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Remark 1.9
  • Remark 1.10
  • ...and 36 more