Relativistic Coulomb problem: lowest-lying energy levels at the critical coupling constant analytically

The Hamiltonian of the spinless relativistic Coulomb problem combines the standard Coulomb interaction potential with the square-root operator of relativistic kinematics. This Hamiltonian is known to be bounded from below up to some well-defined critical coupling constant. At this critical coupling constant, however, the differences between all analytically obtainable upper bounds on the corresponding energy eigenvalues and their numerically determined (approximate) values take their maxima. In view of this, an analytical derivation of (not so bad) upper bounds on the lowest-lying energy levels at the critical coupling constant is presented.