Spectrum of a non-selfadjoint quantum star graph

Gabriel Riviere (1) , Julien Royer (2)

1 LMJL - Laboratoire de Mathématiques Jean Leray
2 IMT - Institut de Mathématiques de Toulouse UMR5219

We study the spectrum of a quantum star graph with a non-selfadjoint Robin condition at the central vertex. We first prove that, in the high frequency limit, the spectrum of the Robin Laplacian is close to the usual spectrum corresponding to the Kirchhoff condition. Then, we describe more precisely the asymptotics of the difference in terms of the Barra-Gaspard measure of the graph. This measure depends on the arithmetic properties of the lengths of the edges. As a by-product, this analysis provides a Weyl Law for non-selfadjoint quantum star graphs and it gives the asymptotic behaviour of the imaginary parts of the eigenvalues.

Domaines: Théorie spectrale [math.SP]; Physique mathématique [math-ph]

Gabriel Riviere, Julien Royer. Spectrum of a non-selfadjoint quantum star graph. 2019. ⟨hal-02356746v2⟩