We study relations between the ground-state energy of a quantum graph Hamiltonian with attractive δ 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 energy increase and decrease are possible for graphs with a more complicated topology.

Abstract. We consider a generalized Schrödinger operator in L 2 (R 2 ) with an attractive strongly singular interaction of δ ′ type characterized by the coupling parameter β > 0 and supported by a C 4 -smooth closed curve Γ of length L without self-intersections. It is shown that in the strong coupling limit, β → 0 + , the number of eigenvalues behaves as Strong δ ′ interaction on a planar loop 2

We provide new estimates on the best constant of the Lieb–Thirring inequality for the sum of the negative eigenvalues of Schrödinger operators, which significantly improve the so far existing bounds.

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