Abstract
AbstractWe prove that every metric graph which is a tree has an orthonormal sequence of generic Laplace-eigenfunctions, that are eigenfunctions corresponding to eigenvalues of multiplicity one and which have full support. This implies that the number of nodal domains $$\nu _n$$
ν
n
of the n-th eigenfunction of the Laplacian with standard conditions satisfies $$\nu _n/n \rightarrow 1$$
ν
n
/
n
→
1
along a subsequence and has previously only been known in special cases such as mutually rationally dependent or rationally independent side lengths. It shows in particular that the Pleijel nodal domain asymptotics from two- or higher dimensional domains cannot occur on these graphs: Despite their more complicated topology, they still behave as in the one-dimensional case. We prove an analogous result on general metric graphs under the condition that they have at least one Dirichlet vertex. Furthermore, we generalize our results to Delta vertex conditions and to edgewise constant potentials. The main technical contribution is a new expression for a secular function in which modifications to the graph, to vertex conditions, and to the potential are particularly easy to understand.
Funder
deutsche forschungsgemeinschaft
Publisher
Springer Science and Business Media LLC
Subject
Mathematical Physics,Statistical and Nonlinear Physics
Cited by
3 articles.
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