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

Pavel Exner, Michal Jex

Published 2011-10-09, updated 2011-12-20Version 2

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.