On the ground state of quantum graphs with attractive $\delta$-coupling

TL;DR

This paper investigates how the ground-state energy of quantum graphs with attractive delta couplings depends on the graph's geometry, revealing conditions under which energy increases with edge length and how topology influences this behavior.

Contribution

It provides a necessary and sufficient condition for energy increase with edge length and explores the impact of graph topology on this relationship.

Findings

Energy increases with edge length in non-branching graphs.

Topology affects the sign change of energy response.

Conditions for energy behavior are explicitly characterized.

Abstract

We study relations between the ground-state energy of a quantum graph Hamiltonian with attractive $\delta$ coupling at the vertices and the graph geometry. We derive a necessary and sufficient condition under which the energy increases with the increase of graph edge lengths. We show that this is always the case if the graph has no branchings while both change signs are possible for graphs with a more complicated topology.