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

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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 (r) whose asymptotic value () is the desired phase shift. A nonlinear first-order ordinary differential equation is derived for (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. ©1963 The American Institute of Physics