\int_{-1}^1 W(t)\, {\mathrm{d}}t$.

One has to compare the resolvents, that of $H_{\alpha, \Gamma}$ given below and the Birman-Schwinger expression of $\left(H_{\epsilon}(W,\gamma) \!-\!k^2
\right)^{-1}$. Both are explicit integral operators and their difference can be treated in a way similar to that used in the squeezing approximation of the one-dimensional $\delta$ interaction — see, e.g., [@AGHH] — a full account of the argument can be found in [@EI].

Notice that the regular potential approximation is not the only way how to justify the leaky-graph model physically; for an alternative see [@FK2].

The resolvent
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As usual, the spectral and scattering properties are encoded in the resolvent and our first task is to find an explicit expression for this operator. We will employ an analogue of the Birman-Schwinger formula for our singular case. If $k^2$ belongs to the resolvent set of $H_{\alpha,\Gamma}$ we put $R^k_{\alpha,\Gamma} := (H_{\alpha,\Gamma}-k^2)^{-1}$. We look for the difference of this operator and the free resolvent $R^k_0$ which is for ${\mathrm{Im\,}}k>0$ an integral operator with the kernel $$\label{freeG}
G_k(x\!-\!y) = {i\over 4}\, H_0^{(1)} (k|x\!-\!y|)\,.$$ To this aim we need embedding operators associated with $R^k_0$. Let $\mu, \nu$ be arbitrary positive Radon measures on ${\mathbb{R}}^2$ with $\mu(x)= \nu(x) =0$ for any 