On Pleijel's nodal domain theorem for quantum graphs

TL;DR

This paper extends Pleijel's theorem to a broad class of quantum graphs, characterizing the asymptotic behavior of nodal domains of eigenfunctions without restrictions on graph structure or edge lengths.

Contribution

It establishes metric graph counterparts of Pleijel's theorem, describing the accumulation points of the normalized nodal domain count for various operators and conditions.

Findings

Accumulation points of normalized nodal domains form a finite subset of (0,1]

Results apply to Schrödinger operators with $L^1$-potentials and the p-Laplacian

In certain cases, the nodal count does not asymptotically equal n

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 Schr\"odinger 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 ${\nu_n}\not\sim {n}$; but in this case even the set of points of accumulation may depend on the choice of eigenbasis.