New Class of Three-Body States

We calculate the three-body spectrum for identical bosons interacting via attractive $1/r^2$ potentials. We find an infinite number of three-body states even when the pair interactions are too weak to support any two-body states. These new states thus share this surprising scenario with the Efimov effect, but are not themselves Efimov states. Our effect occurs for both identical bosons and identical fermions, and it persists in the presence of two-body bound states.

Figure 1

(a) The three-body effective potentials $W_{\nu \nu }(R)$ for three identical bosons with ${\alpha }^2=0$. (b) The asymptotic behavior of $W_{00}(R)$ with the indicated values of ${\alpha }^2$. The symbols are solutions of Eq. (3), while the solid lines are fits based on Eq. (9).

Figure 2

(a) Comparison of the three-boson bound state energies calculated from Eq. (2) (circles) and from Eq. (10) (triangles). The lines are added to guide the eye. (b) Fits of the potential parameters $\beta$ and $\delta$ from Eq. (9): symbols denote individual fits of Eq. (9) to the numerical $W_{00}$; and solid lines, fits of Eq. (11) to these values.

Figure 3

(a) The lowest four three-body energies calculated from Eq. (2) for $W_{00}$ from Eq. (8) with a hard wall added at $R=100r_0$. The dashed line denotes the lowest two-body threshold. (b) Schematic three-body binding energies assuming there is a deeply-bound two-body state whose energy is independent of $\alpha$.