The Lagrange-mesh method is known to be an efficient tool for evaluating the bound states of various three-body atomic and molecular systems. By combining it with the complex scaling method, resonances can also be studied. In this paper, this approach is used for evaluating several S-wave resonances of the helium atom and of the negative positronium ion in vacuum and in Debye plasmas. In spite of its simplicity, the Lagrange-mesh method provides resonance energies and widths more accurate than the best literature results.

The resonance energies and widths of the P-wave states of helium and of the positronium ion are studied under the hypothesis that their constituting particles interact trough pure or screened Coulomb forces. The resonances are investigated by combining the Lagrange-mesh method and the complex scaling method. This approach is proved to be efficient: rather simple to implement, fast, and accurate. The obtained results improve by several orders of magnitude the best literature results.