Abstract
A method based on the modified Faddeev equations in the total-angular-momentum representation is developed to treat the zero-energy scattering in symmetric three-body Coulomb systems. The method is applied to calculate the s-wave scattering lengths of the systems ${\mathit{e}}^{\mathrm{\ensuremath{-}}}$+(${\mathit{e}}^{\mathrm{\ensuremath{-}}}$${\mathit{e}}^{+}$), p+(p\ensuremath{\mu}), and d+(d\ensuremath{\mu}). A large negative value of the even-parity p+(p\ensuremath{\mu}) scattering length indicates the presence of a virtual state near threshold.
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