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Öğe NEW APPROACH TO THE DETERMINATION OF EIGENVALUES AND EIGENFUNCTIONS FOR A RELATIVISTIC 2-FERMION EQUATION(Amer Inst Physics, 1993) BARUT, AO; BRACKEN, AJ; KOMY, S; UNAL, NWorking directly from a covariant equation for two interacting fermions with equal masses and opposite charges, energy eigenvalues and eigenfunctions are calculated to order alpha6 and alpha4, respectively, where a is the fine structure constant. The eigenvalues agree with those determined previously by perturbative techniques, including relativistic, recoil and spin corrections, for all the energy levels of orthopositronium. The eigenvalue problems that arise in the present approach are of Sturm-Liouville-type, but involve one, or two coupled, second-order ordinary differential equations in the radial variable r, with up to four singular points. The method used has two key steps. For values of r of the order of the Bohr radius, the equations are transformed by suitable changes of variable into equations which are equivalent, to the appropriate order of approximation, but which are exactly soluble. The second step involves adjusting the behavior at r=0 of the resultant solutions to match the known behavior of the (unknown) exact eigenfunctions. This approach enables approximate eigenvalues to be determined directly, and corresponding approximate eigenfunctions to be obtained for the first time in simple closed form.
