by $$\label{freeG3}
G_k(x\!-\!y) = \frac{{\mathrm{e}}^{ik|x-y|}} {4\pi \left| x\!-\!y\right|}\,.$$ However, the use of Posilicano’s abstract result requires to interpret the embedding operators involved not as maps between $L^2$ spaces, but rather the last factor $R_{\Gamma}^k$ as $W^{2,2}({\mathbb{R}}^3)\to L^2({\mathbb{R}})$ and its counterpart $\breve{R}_{\Gamma}^k$ as the Banach space dual to $\overline{R_{\Gamma}^k}$. Then $$\label{3BS}
 R^k= R_0^k
 -\breve{R}_{\Gamma}^k (Q^k\!-\!\alpha )^{-1} R_{\Gamma}^k\,,$$ where the modified position of $\alpha$ in this formula corresponds to the usual convention about the coupling parameter for two-dimensional point interactions [@AGHH] reflected in (\[cod2bc\]); roughly speaking, it is an inverse of the one appearing in (\[2-1bc\]). The operator $Q^k$ is the counterpart to $R^k_{m,m}$ of Proposition \[BS\] but we use on purpose a different symbol to stress that we cannot write it simply as an integral operator and a renormalization is needed, cf. [@EK2] for more details.

Geometrically induced properties {#s: geom}
================================

We said in the opening that even if we think about $\Gamma$ in the usual quantum graph model as embedded in ${\mathbb{R}}^\nu$, the shapes of the edges do not influence the spectrum. Leaky graphs are different as one can illustrate in various ways.

Bound states due to non-straightness {#ss:
nonstr}
------------------------------------

Consider again a leaky graph in ${\mathbb{R}}^2$. If 