graph Hamiltonians
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Quadratic forms and boundary conditions {#ss:
qf&bc}
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The Hamiltonians we are interested in are generalized Schrödinger operators with a singular interaction supported by a graph-like $\Gamma$ which is a zero measure set in ${\mathbb{R}}^\nu$. We will use facts about such operators derived, in particular, in [@BEKS] specifying them to our present purpose. Let us first suppose that the configuration space dimension $\nu=2$ and the coupling “strength” is constant on the interaction support.

To begin with, let us show how such a singular operator can be defined generally through the associated quadratic form. Consider a positive Radon measure $m$ on ${\mathbb{R}}^2$ and a number $\alpha>0$ such that $$\label{basiccond}
(1+\alpha) \int_{{\mathbb{R}}^2} |\psi(x)|^2\, {\mathrm{d}}m(x) \le a \int_{{\mathbb{R}}^2}
|\nabla\psi(x)|^2\, {\mathrm{d}}x + b \int_{{\mathbb{R}}^2} |\psi(x)|^2\, {\mathrm{d}}x$$ holds for all $\psi\in{\mathcal{S}}({\mathbb{R}}^2)$ and some $a<1$ and $b$. The map $I_m$ defined by $I_m\psi=\psi$ on the Schwartz space ${\mathcal{S}}({\mathbb{R}}^2)$ extends by density uniquely to $$I_m:\: W^{1,2}({\mathbb{R}}^2) \,\to\, L^2(m):= L^2({\mathbb{R}}^2,m) \;;$$ for brevity the same symbol is used for a continuous function and the corresponding equivalence classes in both $L^2({\mathbb{R}}^2)$ and $L^2(m)$. The inequality (\[basiccond\]) extends to $W^{1,2}({\mathbb{R}}^2)$ with $\psi$ replaced by $I_m\psi$ at the left-hand side. The quadratic form $$\label{Hamform}
{\mathcal{E}}_{-\alpha m}[\psi] :=  \int_{{\mathbb{R}}^2} 