Spectrum of a non-selfadjoint quantum star graph

Abstract

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.

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Cited by in corpus (7)

• Differences between Robin and Neumann eigenvalues
• Comparing the spectrum of Schrödinger operators on quantum graphs
• Differences between Robin and Neumann eigenvalues on metric graphs
• On the eigenvalues of quantum graph Laplacians with large complex $\delta$ couplings
• Some spectral comparison results on infinite quantum graphs
• A modified local Weyl law and spectral comparison results for ${\delta }^{\prime }$-coupling conditions
• Scattering resonances of large weakly open quantum graphs