Accumulation of Three-Body Resonances above Two-Body Thresholds

We calculate resonances in the three-body system with attractive Coulomb potential by solving homogeneous Faddeev-Merkuriev integral equations for complex energies. The equations are solved using the Coulomb-Sturmian separable expansion method. This approach allows us to study the exact behavior of the three-body Coulomb systems near the threshold. A negatively charged positronium ion is used as a test case. In addition to locating all previously known $S$-wave resonances of the positronium ion, we also find a large number of new resonant states that accumulate just slightly above the two-body thresholds. The pattern of accumulation of resonant states above the two-body thresholds suggests that probably they are infinite in number. We conjecture that this may be a general property of the three-body system with an attractive Coulomb potential.

Figure 1

The short-range and long-range potentials, $v^{(s)}$ and $v^{(l)}$, for an attractive Coulomb potential.

Figure 2

Analytic structure of $g_{x_1}(E-\mathrm{i}\epsilon -z^{\prime })g_{y_1}(z^{\prime })$ as a function of $z^{\prime }$, $\epsilon >0$. The Green’s operator $g_{y_1}(z^{\prime })$ has a branch cut on the $[0,\infty )$ interval, while $g_{x_1}(E-\mathrm{i}\epsilon -z^{\prime })$ has a branch cut on the $(-\infty ,E-\mathrm{i}\epsilon ]$ interval and infinitely many poles accumulated at $E-\mathrm{i}\epsilon$ (denoted by dots). The contour $C$ encircles the branch cut of $g_{y_1}$ such that a part of it goes on the unphysical Riemann sheet of $g_{y_1}$ (indicated by the dotted line) and the other part detours away from the cut. The branch cut and some of the poles of $g_{x_1}$ (depicted by dots) are lying on the physical Riemann-sheet. Other poles (depicted by circles) are lying on the unphysical Riemann-sheet of $g_{y_1}$. The contour as shown avoids the singularities of $g_{x_1}$.

Figure 3

Accumulation of resonances above the two-body thresholds. The energies are expressed in atomic units and measured from the three-body breakup threshold.