  For the pair $\{-\Delta, -\Delta-\alpha\delta_\mathcal{C}\}$ of self-adjoint
Schr\"{o}dinger operators in $L^2(\mathbb{R}^n)$ a spectral shift function is
determined in an explicit form with the help of (energy parameter dependent)
Dirichlet-to-Neumann maps. Here $\delta_{\cal{C}}$ denotes a singular
$\delta$-potential which is supported on a smooth compact hypersurface
$\mathcal{C}\subset\mathbb{R}^n$ and $\alpha$ is a real-valued function on
$\mathcal{C}$.
