Evaluation of relativistic transition matrix elements and their contributions to the discrete-spectrum Lamb shift in hydrogenic atoms

For the first time to our knowledge, a general formula for exact transition matrix elements relativistic hydrogenic atoms is derived. By using the obtained formula the matrix elements for different electric and magnetic multipole transitins are evaluated analytically and numerically. Finally, by applying the exact transition matrix elements the contributions of individual discrete states and those of the whole discrete spectrum to the Lamb shift of the 1s state are explicitly calculated.