Schr\"{o}dinger operators on star graphs with singularly scaled potentials supported near the vertices

We study Schr"{o}dinger operators on star metric graphs with potentials of the form $\alpha {\epsilon }^{-2}Q({\epsilon }^{-1}x)$. In dimension 1 such potentials, with additional assumptions on $Q$, approximate in the sense of distributions as $\epsilon \to 0$ the first derivative of the Dirac delta-function. We establish the convergence of the Schr"{o}dinger operators in the uniform resolvent topology and show that the limit operator depends on $\alpha$ and $Q$ in a very nontrivial way.