Singular quasipotential equation

A study is made of the quasipotential equation for the partial-wave scattering amplitude in momentum space. For singular quasipotentials $V(r)=gr^{-2n+1}$ ($n$ integral, greater than or equal to 1) the integral equation reduces to an inhomogeneous differential equation of order $2n$ with definite boundary conditions. For $n=2$, $l>0$, the existence and uniqueness of the solution of the corresponding boundary-value problem is proved. It is proposed to construct the solution in the $S$-wave case ($l=0$) by analytic continuation in $l$. It is shown that the solution obtained in this manner satisfies an integral equation with a potential that differs from the analytic continuation in $l$ of the original polynomial by a definite polynomial. The solutions that are found can be represented as series in powers of $g^\nu(\ln g)^{n_\nu}$ (modified perturbation theory). An approximate method of investigating quasipotentials with arbitrary (nonintegral) $n$ is proposed.