Weakly coupled states on branching graphs

of N links joined at a single point. Each link supports a real locally integrable potential Vj ; the self–adjointness is ensured by the δ type boundary condition at the vertex. If all the links are semiinfinite and ideally coupled, the potential decays as x−1−ǫ along each of them, is non–repulsive in the mean and weak enough, the corresponding Schrödinger 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 δ coupling constant may be interpreted in terms of a family of squeezed potentials. Recent progress