A spectral shift function for Schr\"{o}dinger operators with singular interactions

For the pair $\{-\mathrm{\Delta },-\mathrm{\Delta }-\alpha {\delta }_{\mathcal{C}}\}$ 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 ${\delta }_{\mathcal{C}}$ 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 $\mathcal{C}$.