obtained by verifying the Birman-Schwinger (BS) formula in our singular setting. The procedure requires some care; a full account concerning the claims (i)–(iii) can be found in [@BEKS], for (iv) see [@Pos2].

The case of codimension two {#ss: cod2}
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The second one of the above constructions of $H_{\alpha,\Gamma}$ can also be rephrased in the following way: we first restrict the Laplacian to a symmetric operator defined on function which vanish in the vicinity of $\Gamma$, and afterwards we choose a particular self-adjoint extension specified by the condition (\[2-1bc\]). It follows from general properties of partial differential operators [@He] that a similar construction is possible also in higher dimensions as long as $\mathrm{codim\,}\Gamma\le 3$. Since we want to stick to cases of physical interest, we will mention here only graphs whose edges are curves in ${\mathbb{R}}^3$.

An analogue of the form definition (\[Hamform\]) does not work in this situation and we have to rely on boundary conditions which are, however, more complicated than (\[2-1bc\]). The difference is of a local character, thus we restrict ourself to the simplest situation when $\Gamma$ is a single infinite curve described by a $C^2$ function $\gamma (s):\:{\mathbb{R}}\to {\mathbb{R}}^{3}$ without self-intersections and such that $|\dot 