Klein-Gordon equation on a Lagrange mesh

The Lagrange-mesh method is an approximate variational method which provides accurate solutions of the Schrödinger equation for bound-state and scattering few-body problems. The stationary Klein-Gordon equation depends quadratically on the energy. For a central potential, it is solved on a Lagrange-Laguerre mesh by iteration. Results are tested with the Coulomb potential for which exact solutions are available. A high accuracy is obtained with a rather small number of mesh points. For various potentials and levels, few iterations provide accurate energies and mean values in short computer times. Analytical expressions of the wave functions are available.

Figure 1

Ground-state wave functions $u_{00}\left(r\right)$ of the Yukawa potential from the Klein-Gordon equation (full line) and from the Schrödinger equation (dashed line).