NEW APPROACH TO THE DETERMINATION OF EIGENVALUES AND EIGENFUNCTIONS FOR A RELATIVISTIC 2-FERMION EQUATION

Working directly from a covariant equation for two interacting fermion s with equal masses and opposite charges, energy eigenvalues and eigen functions are calculated to order alpha6 and alpha4, respectively, whe re a is the fine structure constant. The eigenvalues agree with those determined previously by perturbative techniques, including relativist ic, recoil and spin corrections, for all the energy levels of orthopos itronium. The eigenvalue problems that arise in the present approach a re of Sturm-Liouville-type, but involve one, or two coupled, second-or der 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 b y 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 result ant solutions to match the known behavior of the (unknown) exact eigen functions. This approach enables approximate eigenvalues to be determi ned directly, and corresponding approximate eigenfunctions to be obtai ned for the first time in simple closed form.