Optimization of quantum graph eigenvalues with preferred orientation vertex conditions

TL;DR

This paper investigates how vertex conditions that break time-reversal symmetry affect the Laplacian spectrum on finite metric graphs, identifying optimal configurations for maximizing ground state eigenvalues and providing bounds for all eigenvalues.

Contribution

It introduces methods to optimize eigenvalues of quantum graphs with non-reversible vertex conditions, especially for star graphs and general finite graphs.

Findings

Maximized ground state eigenvalue configurations for star graphs.

Provided upper bounds for all eigenvalues of finite metric graphs.

Identified the impact of non-reversible vertex conditions on the spectrum.

Abstract

We discuss Laplacian spectrum on a finite metric graph with vertex couplings violating the time-reversal invariance. For the class of star graphs we determine, under the condition of a fixed total edge length, the configurations for which the ground state eigenvalue is maximized. Furthermore, for general finite metric graphs we provide upper bounds for all eigenvalues.