In this paper we consider the two-dimensional Schr\”odinger operator with an
attractive potential which is a multiple of the characteristic function of an
unbounded strip-shaped region, whose thickness is varying and is determined by
the function $\mathbb{R}\ni x \mapsto d+\varepsilon f(x)$, where $d > 0$ is a
constant, $\varepsilon > 0$ is a small parameter, and $f$ is a compactly
supported continuous function. We prove that if $\int_{\mathbb{R}} f
\,\mathsf{d} x > 0$, then the respective Schr\”odinger operator has a unique
simple eigenvalue below the threshold of the essential spectrum for all
sufficiently small $\varepsilon >0$ and we obtain the asymptotic expansion of
this eigenvalue in the regime $\varepsilon\rightarrow 0$. An asymptotic
expansion of the respective eigenfunction as $\varepsilon\rightarrow 0$ is also
obtained. In the case that $\int_{\mathbb{R}} f \,\mathsf{d} x < 0$ we prove
that the discrete spectrum is empty for all sufficiently small $\varepsilon >
0$. In the critical case $\int_{\mathbb{R}} f \,\mathsf{d} x = 0$, we derive a
sufficient condition for the existence of a unique bound state for all
sufficiently small $\varepsilon > 0$.



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