Spectrum of a non-selfadjoint quantum star graph

TL;DR

This paper analyzes the spectrum of a non-selfadjoint quantum star graph with Robin boundary conditions, revealing asymptotic behaviors and a Weyl Law influenced by edge lengths and arithmetic properties.

Contribution

It provides a detailed asymptotic analysis of the spectrum for non-selfadjoint quantum star graphs, including a Weyl Law and eigenvalue imaginary part behavior.

Findings

Spectrum asymptotics depend on edge length arithmetic properties

High-frequency spectrum approaches Kirchhoff spectrum

Established a Weyl Law for non-selfadjoint quantum graphs

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.