Weakly coupled states on branching graphs

We consider a Schrodinger 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 semi-infinite and ideally coupled, the potential decays as x(-1-epsilon) along each of them, is nonrepulsive in the mean and weak enough, the corresponding Schrodinger 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.