Analytical computation of amplification of coupling in relativistic equations with Yukawa potential

The approximate analytic solutions to the Klein-Gordon and Dirac equations with the Yukawa potential were derived by using the quasilinearization method (QLM). The accurate analytic expressions for the ground state energies and wave functions were presented. These high-precision approximate analytic representations are obtained by first casting the proper relativistic equation into a nonlinear Riccati form and then solving that nonlinear equation in the first QLM iteration. The choice of zero iteration is based on general features of the exact solutions near the origin and infinity. To estimate the accuracy of the QLM solutions, the exact numerical solutions were found, as well. The analytical QLM solutions are found to be extremely accurate for a small exponent parameter w of the Yukawa potential. The reasonable accuracy is kept for the medium values of w. When w approaches the critical values, the precision of the QLM results falls down markedly. However, the approximate analytic QLM solution to the Dirac equation corresponding to the maximum relativistic effect turned out to be very accurate even for w close to the exact critical wexDir=1.6767, whereas the QLM calculations yield wqlmDir=1.6763. This effect of “amplification” in compare with the Schrödinger equation critical parameter wexSch=1.1906 was investigated earlier [S. De Leo, P. Rotelli, Phys. Rev. D 69 (2004) 034006]. In this work, it was found that the “amplification” for the Klein-Gordon equation became all the more evident. The exact numerical value is wexKG≃2.25, whereas the QLM approximation yields wqlmKG≃2.15.