On Pleijel's nodal domain theorem for quantum graphs

Abstract

We establish metric graph counterparts of Pleijel's theorem on the asymptotics of the number of nodal domains ${\nu }_n$ of the $n$-th eigenfunction(s) of a broad class of operators on compact metric graphs, including Schrödinger operators with $L^1$-potentials and a variety of vertex conditions as well as the $p$-Laplacian with natural vertex conditions, and without any assumptions on the lengths of the edges, the topology of the graph, or the behaviour of the eigenfunctions at the vertices. {Among other things, these results characterise the accumulation points of the sequence $(\frac{{\nu }_n}{n}{)}_{n\in \mathbb{N}}$, which are shown always to form a finite subset of $(0,1]$. This} extends the previously known result that ${\nu }_n\sim n$ \textit{generically}, for certain realisations of the Laplacian, in several directions. In particular, in the special cases of the Laplacian with natural conditions, we show that for graphs with rationally dependent edge lengths, one can find eigenfunctions thereon for which ${\nu }_n\nsim n$; but in this case even the set of points of accumulation may depend on the choice of eigenbasis.

References in corpus (12)

• On the Spectral Gap of a Quantum Graph
• Surgery principles for the spectral analysis of quantum graphs
• Gaussian estimates for a heat equation on a network
• Nodal domains on quantum graphs
• On the connection between the number of nodal domains on quantum graphs and the stability of graph partitions
• A lower bound for nodal count on discrete and metric graphs
• Edge connectivity and the spectral gap of combinatorial and quantum graphs
• The Nodal Count {0, 1, 2, 3,...} Implies The Graph is a Tree
• Nodal Statistics On Quantum Graphs
• The first eigenvalue of the $p-$Laplacian on quantum graphs
• Stability of eigenvalues of quantum graphs with respect to magnetic perturbation and the nodal count of the eigenfunctions
• On extremal eigenvalues of the graph Laplacian

Cited by in corpus (2)

• Universality of nodal count distribution in large metric graphs
• On fully supported eigenfunctions of quantum graphs