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 $$\nu _n$$ ν n of the nth eigenfunction(s) of a broad class of operators on compact metric graphs, including Schrödinger operators with $$L^1$$ 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 $$(\frac{\nu _n}{n})_{n\in \mathbb {N}}$$ ( ν n n ) n ∈ N , which are shown always to form a finite subset of (0, 1]. This extends the previously known result that $$\nu _n\sim n$$ ν n ∼ n 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 any graph with pairwise commensurable edge lengths and at least one cycle, one can find eigenfunctions thereon for which $${\nu _n}\not \sim {n}$$ ν n ≁ n ; but in this case even the set of points of accumulation may depend on the choice of eigenbasis.

On fully supported eigenfunctions of quantum graphs

“…This has been proved only under technical assumptions so far. Our first result, Theorem 1 proves that this is indeed universally true on tree graphs with standard conditions at all vertices, thus partly settling a conjecture asked in [14]. Theorem 2 generalizes this to all connected graphs with at least one Dirichlet vertex.…”

“…We also emphasize that the assumed connectedness of G refers to connectedness after all Dirichlet vertices have been split and turned into vertices of degree one as just described. Our starting point is Conjecture 4.3 in [14] which states that for every compact connected metric graph there exists an orthonormal base (ψ n ) n∈N of Laplaceeigenfunctions with monotonous eigenvalues such that such that for a subsequence (n k ) k∈N , all ψ k n have full support, i.e. none of these eigenfunctions vanishes identically on any edge of G. Together with Proposition 3.4 in [14], this would imply for this choice of eigenfunctions lim sup…”

“…In fact, there is a remarkable dichotomy of special cases in which the conjecture is known to be true: on the one hand, these are graphs with pairwise rationally dependent side lengths, cf. [14,Theorem 4.2]. The crucial idea is to use their resonance and to construct a fully supported eigenfunction by putting cosine waves on all edges.…”

Doubly nonlinear Schrödinger ground states on metric graphs

“…Example of a graph fulfilling Assumption (H) as in [15]. to [19,22,35,36,39,45] and references therein for some of the most recent developments. In the nonlinear case, a prominent focus has been devoted to Schrödinger equations (see for instance [11, 12, 21, 23, 24, 31-33, 38, 44, 53, 54, 56] as well as the recent review [3] and references therein), but other nonlinear models have been considered too (see [51] for the KdV equation and [26,27] for the Dirac equation).…”

Generic Laplace eigenfunctions on metric graphs

“…We may write it as f | e j ≡ 0 for every edge e j . Conjecture 4.3 in [22] states, Conjecture 7.2 (Full support eigenfunctions). [22] For any metric graph (Γ, ℓ), and any choice of a complete orthonormal sequence of eigenfunctions, there are infinitely many eigenfunctions with full support.…”