Universal low-energy properties of three two-dimensional bosons

Universal low-energy properties are studied for three identical bosons confined in two dimensions. The short-range pairwise interaction in the low-energy limit is described by means of the boundary condition model. The wave function is expanded in a set of eigenfunctions on the hypersphere, and the system of hyperradial equations is used to obtain analytical and numerical results. Within the framework of this method, exact analytical expressions are derived for the eigenpotentials and the coupling terms of hyperradial equations. The derivation of the coupling terms is generally applicable to a variety of three-body problems provided the interaction is described by the boundary condition model. The asymptotic form of the total wave function at a small and a large hyperradius $\rho$ is studied, and the universal logarithmic dependence $\sim {\ln }^3\rho$ in the vicinity of the triple-collision point is derived. Precise three-body binding energies and the $(2+1)$-scattering length are calculated.

Figure 1

Convergence of eigenvalues ${\xi }_n\left(R\right)$ to the limiting function (bold line). Two families of ${\xi }_n\left(R\right)$ are plotted by solid lines for $n=3m$ and by dashed lines for $n=3m+1$, $m=1,5,9,15,25$.

Figure 2

Two families of $Q_{n1}\left(R\right)$ (a) and $P_{n1}\left(R\right)$ (b) are plotted by solid lines for $n=3m$ and by dashed lines for $n=3m+1$, $m=1,5,9,15,25$. The corresponding limiting functions ${\tilde{Q}}_1(R-\ln n-\gamma )$ and ${\tilde{P}}_1(R-\ln n-\gamma )$ are plotted by a bold line. In panel (c) the exact result (solid lines) and the asymptotic expression (42) (dashed lines) are compared for the third family of $Q_{n1}\left(R\right)$ ($n=3m+2$, $m=1,5,9,15,25$).

Figure 3

The lowest scaled eigenpotentials ${\xi }_n^2\left(R\right)e^{-2R}$. Notice different scales for the positive and negative $R$. The arrow marks the two-body bound-state energy $-4e^{-2\gamma }$.

Figure 4

Coupling terms $Q_{mn}\left(R\right)$ (a), $P_{nn}\left(R\right)$ (b), and $P_{nm}\left(R\right)$ for $n\ne m$ (c). The arrow marks the large hyperradius limit $1∕3$ of $P_{11}\left(R\right)$.

Figure 5

Radial functions of the four lowest channels $f_1\left(R\right)$ (a), $f_2\left(R\right)$ (b), $f_3\left(R\right)$ (c), and $f_4\left(R\right)$ (d) for the ground state (solid lines), the excited state (dashed lines), and the scattering state calculated at the two-body threshold energy (dotted lines). For convenience, the radial functions for the excited state are multiplied by a factor of 5, while those for the scattering state are scaled to match at the first maximum the first-channel functions of the excited state and scattering state. Linear asymptotic dependence of the first-channel scattering solution is shown by a thin straight line in panel (a).