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Title:
Generic Laplace eigenfunctions of finite metric graphs
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Abstract:
Up to certain pathologies, a compact metric graph with standard vertex conditions has a Baire-generic set of choices of edge lengths such that all Laplacian eigenvalues are simple (Friedlander 05’) with eigenfunctions that do not vanish at any vertex (Berkolaiko-Liu 17’). However, this set of edge lengths is implicit. An explicit condition can be made, at the expense of a density zero subsequence of eigenvalues. For any choice of rationally-independent edge lengths, almost every eigenvalue is simple and has an eigenfunction that does not vanish on vertices (Alon-Band-Berkolaiko 18’). I will refer to the latter type of genericity as "ergodic genericity." To explain what is the ergodic system in the background, I will introduce a moduli space of solutions (eigenpairs) to all possible choices of edge lengths. I will explain how the eigenpairs of a graph with rationally-independent lengths equidistribute in this moduli space. Recently, Kurasov and Sarnak proved a conjecture of Colin de Verdiere regarding the irreducible algebraic structure of this moduli space. I will show how this irreducibility can be applied to prove the previous genericity results and many more. I will show that, generically, an eigenfunction fails to satisfy any additional vertex condition. Moreover, I will show that given two different metric graphs with the same edge lengths, generically, they do not share any non-zero eigenvalue.
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