through the edge lengths, their shapes being irrelevant

A way to deal with the first problem is to regard a quantum graph as an idealization of a more “realistic” system without such ambiguities; a natural candidate for this role are various “fat graphs”. Limits of such objects when the edge width squeezes to zero were studied extensively, first in the easier Neumann-type case [@FW; @KZ; @RS; @Sa; @Ku2; @KZ2; @EP1; @Po1; @EP2] and quite recently also in the situation with Dirichlet boundaries [@Po2; @MV; @CaE; @Gr]. These results give a partial answer to the first question[^3] while the second problem remains.

Here we are going to discuss a class of quantum graph models which are free of both difficulties; the price we pay is — similarly as for the fat graphs mentioned above — that instead of ordinary differential equations we have to deal with a PDE problem. The idea is to preserve the whole Euclidean space as the configuration space and to suppose that the particle is kept in the vicinity of the graph $\Gamma\subset \mathbb{R}^\nu$ by an attractive singular interaction. Formally such a Hamiltonian expresses as $$\label{formal}
-\Delta -\alpha(x) \delta (x-\Gamma )\,$$ with $\alpha(x)>0\,$; we will consider mostly 