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

Abstract

We study Schr\"{o}dinger operators on star metric graphs with potentials of the form $\alpha\varepsilon^{-2}Q(\varepsilon^{-1}x)$. In dimension 1 such potentials, with additional assumptions on $Q$, approximate in the sense of distributions as $\varepsilon\to0$ 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.