Quantum Graphs: Coulomb-Type Potentials and Exactly Solvable Models

We study the Schrodinger operators on a non-compact star graph with the Coulomb-type potentials having singularities at the vertex. The convergence of regularized Hamiltonians H-epsilon with cutoff Coulomb potentials coupled with (alpha delta+beta delta')-like ones is investigated. The 1D Coulomb potential and the delta ' -potential are very sensitive to their regularization method. The conditions of the norm resolvent convergence of H-epsilon depending on the regularization are established. The limit Hamiltonians give the Schrodinger operators with the Coulomb-type potentials in a mathematically precise meaning, ensuring the correct choice of vertex conditions. We also describe all self-adjoint realizations of the formal Coulomb Hamiltonians on the star graph.