Reduction of quasipotential equations to Sturm–Liouville problems and the comparison equation method

The quasipotential equation that describes the interaction of two scalar particles of equal mass through a quasipotential of the form $V(r)=-gr^{-1}$ in the coordinate space is reduced to the corresponding Sturm–Liouville problem in the momentum space. An iterative comparison equation method is formulated in general form and makes it possible to find the solutions of such problems as asymptotic series in reciprocal powers of the coupling constant. The method is used to obtain the discrete spectrum at all binding energies. The problem of the equality of the solutions of the comparison equation method and the exact solutions at singular points of the original equation is investigated for the example of a modified variant of the Sturm–Liouville quasipotential problem. Such an investigation is needed if the original equation does not admit exact comparison.