Weakly coupled states on branching graphs

We consider a Schr"odinger particle on a graph consisting of $N$ links joined at a single point. Each link supports a real locally integrable potential $V_j$; the self–adjointness is ensured by the $\delta$ type boundary condition at the vertex. If all the links are semiinfinite and ideally coupled, the potential decays as $x^{-1-ϵ}$ along each of them, is non–repulsive in the mean and weak enough, the corresponding Schr"odinger operator has a single negative eigenvalue; we find its asymptotic behavior. We also derive a bound on the number of bound states and explain how the $\delta$ coupling constant may be interpreted in terms of a family of squeezed potentials.