can check that $\dot H_{\alpha,\Gamma}$ is e.s.a., and moreover, by Green’s formula it reproduces the form (\[Hamform\]) on its core, so its closure may be identified with $H_{\alpha,\Gamma}$ defined above.

\[rem1def\] (i) The above definitions easily extend to the situation with the singular interaction strength $\alpha(s)$ varying along the edges provided the corresponding function $\alpha:\Gamma\to{\mathbb{R}}_+$ is sufficiently regular.\
(ii) In a similar way one can define operators corresponding to the formal expression (\[formal\]) for a generalized “graph” whose edges are $(\nu-1)$–dimensional manifolds in ${\mathbb{R}}^\nu$ satisfying suitable regularity conditions analogous to (g1)–(g3).

Regular potential approximation {#ss: regular}
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As we shall see below the operators $H_{\alpha,\Gamma}$ represent a reasonably general class of systems for which various properties can be derived. One can ask nevertheless whether this is not again a too idealized model. Before proceeding further we want to show that $H_{\alpha,\Gamma}$ can be regarded as weak-coupling approximation to a class of regular Schrödinger operators; for simplicity we restrict ourself to graphs with a single infinite edge.

Let $\Gamma$ be a curve described by a $C^2$ function $\gamma:\:{\mathbb{R}}\to{\mathbb{R}}^2$. Then we are able to define the signed curvature $k(s):= \left(\dot\gamma_1 \ddot\gamma_2 - \dot\gamma_2
\ddot\gamma_1 \right)(s)$; we assume that it is bounded, $|k(s)|<c_+$ 