Adiabatic hyperspherical representation for the three-body problem in two dimensions

We explore the three-body problem in two dimensions using the adiabatic hyperspherical representation. We develop the main equations in terms of democratic hyperangular coordinates and determine several symmetry properties and boundary conditions for both interacting and noninteracting solutions. From the analysis of the three-body effective potentials, we determine the threshold laws for low-energy three-body recombination, collision-induced dissociation, as well as inelastic atom-diatom collisions in two dimensions. Our results show that the hyperspherical representation can offer a simple and conceptually clear physical picture for three-body process in two dimensions which is also suitable for calculations using finite-range two-body interactions supporting a number of bound states.

Figure 1

Three-body potentials for three identical bosons in 2D with (a) ${\left|M\right|}_r^{\pi }=0_s^{+}$ (solid lines) and $0_a^{+}$ (dashed lines), (b) ${\left|M\right|}_r^{\pi }=1_s^{-}$ and $1_a^{-}$, and (c) ${\left|M\right|}_r^{\pi }=2_s^{+}$ and $2_a^{+}$ (shaded region indicates the region in $R$ in which particles can be found at distances smaller than the range of the interaction $r_0)$. Note that for $\left|M\right|\ne 0$, the potentials for the symmetric and antisymmetric three-body systems (with respect to reflection) are exactly degenerate, while for $\left|M\right|=0$ they are not. Note also that $m_{2\mathrm{b}}=0$ states are forbidden for the $0_a^{+}$ symmetry in (a).

Figure 2

Three-body potentials for three identical fermions in 2D with (a) ${\left|M\right|}_r^{\pi }=0_s^{+}$ (solid lines) and $0_a^{+}$ (dashed lines), (b) ${\left|M\right|}_r^{\pi }=1_s^{-}$ and $1_a^{-}$, and (c) ${\left|M\right|}_r^{\pi }=2_s^{+}$ and $2_a^{+}$ (shaded region indicates the region $R$ in which particles can be found at distances smaller than $r_0)$. Note that, similar to identical bosons, the $\left|M\right|\ne 0$ potentials for the symmetric and antisymmetric 2D three-body systems (with respect to reflection) are exactly degenerate, while for $\left|M\right|=0$ they are not.

Figure 3

Relation of the coordinate $\theta$ to the relative positions of the Jacobi vectors in the body frame.

Figure 4

Jacobi vectors before (black) and after (red) the parity operation $\Pi$.

Figure 5

Jacobi vectors before (black) and after (red) the $P_{12}$ permutation operation.

Figure 6

Jacobi vectors before (black) and after (red) the $P_{23}$ permutation operation.