  We investigate the Schr\"{o}dinger operators $H_\varepsilon=-\Delta
+W+V_\varepsilon$ in $\mathbb{R}^2$ with the short-range potentials
$V_\varepsilon$ which are localized around a smooth closed curve $\gamma$. The
operators $H_\varepsilon$ can be viewed as an approximation of the heuristic
Hamiltonian $H=-\Delta+W+a\partial_\nu\delta_\gamma+b\delta_\gamma$, where
$\delta_\gamma$ is Dirac's $\delta$-function supported on $\gamma$ and
$\partial_\nu\delta_\gamma$ is its normal derivative on $\gamma$. Assuming that
the operator $-\Delta +W$ has only discrete spectrum, we analyze the asymptotic
behaviour of eigenvalues and eigenfunctions of $H_\varepsilon$. The
transmission conditions on $\gamma$ for the eigenfunctions $u^+=\alpha u^-$,
$\alpha\, \partial_\nu u^+-\partial_\nu u^-=\beta u^-$, which arise in the
limit as $\varepsilon\to 0$, reveal a nontrivial connection between spectral
properties of $H_\varepsilon$ and the geometry of $\gamma$.
