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

摘要

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 expres-sions for the ground state energies and wave functions were presented. These high-precision approximate analytic representa-tions are obtained by first casting the proper relativistic equation into a nonlinear Riccati form and then solving that nonlinear equa-tion in the first QLM iteration. The choice of zero iteration is based on general features of the exact solutions near the origin and infin_ity. To estimate the accuracy of the QLM solutions, the exact numerical solutions were found, as well. The analytical QLM solu_tions 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. How-ever, 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 w_(ex)~(Dir) = 1.6767, whereas the QLM calculations yield w_(qlm)~(Dir) = 1.6763. This effect of "amplification" in compare with the Schr_dinger equation critical parameter w_(ex)~(Sch) = 1.1906 was investigated earlier [S. De Leo, P. Rotelli, Phys. Rev. D 69 (2004) 0340061. In this work, it was found that the "ampliflcation" for the Klein-Gordon equation became all the more evident. The exact numerical value is w_(ex)~(KG) ≈ 2.25, whereas the QLM approximation yields w_(qlm)~(KG)≈2.15.