for some $c_+>0$ and all $s\in{\mathbb{R}}$. Moreover, we suppose that $\Gamma$ has neither self-intersections nor “near-intersections”, i.e. that there is a $c_->0$ such that $|\gamma(s)\!-\!\gamma(s')|\ge c_-$ for any $s,s'$ with $|s\!-\!s'|\ge c_-$. Then we can define in the vicinity of $\Gamma$ the standard locally orthogonal system of coordinates [@ES], i.e. the pairs $(s,u)$ where $u$ is the (signed) normal distance from $\Gamma$ and $s$ is the arc-length coordinate of the point of $\Gamma$ where the normal $n(s)$ is taken; the system is unique in the strip neighborhood $\Sigma_{\epsilon}:= \{ x(s,u):\:
(s,u)\in \Sigma^0_{\epsilon}\}$, where $$\label{curvi}
x(s,u):= \gamma(s) + n(s)u$$ and $\Sigma^0_{\epsilon}:= \{(s,u):\: s\in{\mathbb{R}}, |u|<\epsilon\}$ as long as the condition $2\epsilon< c_-$ is valid.

With these prerequisites we can construct the approximating family. Given $W\in L^{\infty}((-1,1))$, we define for all $\epsilon< {1\over 2}\, c_-$ the transversally scaled potential, $$\label{scaledpot}
V_{\epsilon}(x) := \left\lbrace
\begin{array}{ccl} 0 & \quad\dots\quad & x\not\in\Sigma_{\epsilon}
\\ [.5em] -\,{1\over\epsilon}\, W\left(u\over\epsilon \right) &
\quad\dots\quad & x\in\Sigma_{\epsilon} \end{array} \right.$$ and put $$\label{scaledHam}
H_{\epsilon}(W,\gamma):= -\Delta\, +\, V_{\epsilon} \,.$$ The operators $H_{\epsilon}(W,\gamma)$ are obviously self-adjoint on $D(-\Delta)= W^{2,2}({\mathbb{R}}^2)$ and we have the following approximation result:

\[ditch\_approx\] Under the stated assumptions, $H_{\epsilon}(W,\Gamma) \to
H_{\alpha, \Gamma}$ as $\epsilon\to 0$ in the norm-resolvent sense, where $\alpha := 