Atomic three-body resonances

Two methods of locating resonances of the e -- Ps (e- -- e - -- e+) atomic three-body system are presented. In the first method the homogeneous Faddeev-Merkuriev integral equations are solved by applying a separable expansion approximation on the potential terms in the Coulomb-Sturmian basis. This approximation transforms the integral equations into a matrix equation. The Coulomb-Sturmian matrix elements of the three-body Coulomb Green's operator are then calculated as a contour integral of the two-body Coulomb's Green matrices. The calculation of the Coulomb Green's matrices are considerably simplified by the use of continued fractions that result from a tri-diagonal Jacobi matrix. The complex energies are searched for as the complex zeros of the Fredholm determinant. In the second method the scattering matrix S is computed by means of resolvent Green's operators. The eigenphase sum is then calculated as a function of energy and the resonances appear as singularities of the plot of the eigenphase sum formula.