Bound state calculation of two-component Klein-Gordon equation with Damped-Relaxation technique

The Damped-Relaxation method is applied to the two-component Klein-Gordon equation to find the bound state of the electron in the external scalar field. With the finite-difference representation of the differential operators, one-, two- and three-dimensional problems are solved to obtain the ground-state spinor of the equation. The convergence with respect to the damping parameters is analyzed. As references, the solution by the real-time propagation method as well as the wave function of the non-relativistic equation examine the present results.