Hiatus perturbation for a singular Schr\"odinger operator with an interaction supported by a curve in $\mathbb{R}$^3

We consider Schr"odinger operators in $L^2({\mathbb{R}}^3)$ with a singular interaction supported by a finite curve $\mathrm{\Gamma }$. We present a proper definition of the operators and study their properties, in particular, we show that the discrete spectrum can be empty if $\mathrm{\Gamma }$ is short enough. If it is not the case, we investigate properties of the eigenvalues in the situation when the curve has a hiatus of length $2ϵ$. We derive an asymptotic expansion with the leading term which a multiple of $ϵ\ln ϵ$.