Low-Energy Expansion of Scattering Phase Shifts for Long-Range Potentials

Abstract

Phase shifts can be used to describe the quantum mechanical scattering of a particle by a spherically symmetric potential. They are odd functions of the wave number k, which is proportional to the square root of the energy of the incident particle. For short‐range potentials, they are analytic at k = 0 and can be expanded in odd powers of k. However for long‐range potentials they are not analytic at k = 0 so they cannot be expanded in powers of k. Since electron‐atom, atom‐atom, proton‐neutron, and multipole‐multipole potentials are of long range, it is important to consider such potentials. A method is presented for determining the appropriate expansions around k = 0. The method is applied to potentials which are O(r^−v) as r → ∞, and the phase shifts for any angular momentum are obtained up to and including the first nonanalytic term. For L = 0 and 3 < v < 4 the next term is also obtained. The nonanalytic terms involve ln k and fractional powers of k. The results for v = 4 and the k dependence for v = 2L + 3 agree with those obtained in a different way by O'Malley, Spruch, and Rosenberg. The method involves a function η(r) whose asymptotic value η(∞) is the desired phase shift. A nonlinear first‐order ordinary differential equation is derived for η(r). This equation is solved by expansion and iteration methods. To expand the solution around k = 0, two simple theorems are proved concerning the asymptotic forms of certain integrals containing a parameter.