On Pleijel\u2019s 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 nth 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})_{nin mathbb {N}}, which are shown always to form a finite subset of (0, 1]. This extends the previously known result that nu _nsim ngenerically, 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 any graph with pairwise commensurable edge lengths and at least one cycle, 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.