Eigenchannel method for nonrelativistic scattering from zero-range potentials

An eigenchannel method for nonrelativistic quantum scattering from an arbitrary system of zero-range potentials is presented. Eigenchannel vectors are defined as characteristic vectors of an auxiliary Hermitian matrix spectral problem with a positive semidefinite weight; real eigenvalues to this problem are the negative cotangents of eigenphase shifts. The eigenchannel vectors and the eigenphase shifts are used to construct particular solutions, called eigenchannels, of the pertinent Schrodinger equation. Analysis of the asymptotics of the eigenchannels in the far zone leads to a definition of eigenchannel harmonics, which for the problem at hand play the role analogous to that played by spherical harmonics for scattering in a central field. Representations of a far-field amplitude (for scattering of particles emitted from a point source) as well as a scattering amplitude, a scattering kernel, total, and total averaged cross sections (for scattering of a parallel beam of projectiles) in terms of the eigenphase shifts and the eigenchannel harmonics are derived. As an illustrative example, scattering of a plane wave from a system of four identical zero-range potentials, located in vertices of a regular tetrahedron, is worked out in detail.