arXiv subjects
Publications and source records attributed to Duc Hoang Cao.
We explore a connection between geodesic nets and quantum graphs optimising certain functionals from spectral theory. For surfaces, critical metrics for the normalised $k^{\mathrm{th}}$ eigenvalue of the Laplacian give rise to isometric minimal immersions to a unit sphere. In this spirit we obtain geodesic nets from optimal quantum graphs, and obstructions to the existence of critical metrics.
We prove sharp upper bounds for eigenvalues of Schrödinger operators on quantum graphs with $δ$-coupling (also known as Robin) conditions at all vertices. The bounds depend on the geometry of the graph, on the potential, and the strength of the couplings, and as the coupling strengths grow, the dependence on the topology gets weaker, answering a question of Rohleder and Seifert. We obtain those bounds via the variational characterisation, comparing with appropriate linear combinations of eigenfunctions with Dirichlet and Neumann vertex conditions.