Differences Between Robin and Neumann Eigenvalues on Metric Graphs

Ram Band, Holger Schanz, Gilad Sofer

Research output: Contribution to journalArticlepeer-review

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 by Rudnick et al. (Commun Math Phys, 2021. arXiv:2008.07400). 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.

Original languageEnglish
Pages (from-to)3859-3898
Number of pages40
JournalAnnales Henri Poincare
Volume25
Issue number8
DOIs
StateAccepted/In press - 2023

All Science Journal Classification (ASJC) codes

  • Nuclear and High Energy Physics
  • Statistical and Nonlinear Physics
  • Mathematical Physics

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