Differences between Robin and Neumann eigenvalues on metric graphs

TL;DR

This paper investigates the differences between Robin and Neumann eigenvalues on metric graphs, establishing their limiting behavior, bounds, and relation to geometric properties through spectral analysis techniques.

Contribution

It introduces the concept of the Robin-Neumann gap on metric graphs and proves the existence of its mean value, bounds, and connections to geometric and probabilistic aspects.

Findings

The Robin-Neumann gap sequence has a well-defined mean value equal to a geometric quantity.

The sequence of gaps is uniformly bounded with explicit bounds.

The study links the accumulation points of gaps to their probability distribution.

Abstract

We consider the Laplacian on a metric graph, equipped with Robin ($\delta$-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.