Universal description of the rotational-vibrational spectrum of three particles with zero-range interactions

A comprehensive universal description of the rotational-vibrational spectrum for two identical particles of mass $m$ and the third particle of the mass $m_1$ in the zero-range limit of the interaction between different particles is given for arbitrary values of the mass ratio $m/m_1$ and the total angular momentum $L$. If the two-body scattering length is positive, a number of vibrational states is finite for $L_c(m/m_1) \le L \le L_b(m/m_1)$, zero for $L>L_b(m/m_1)$, and infinite for $L<L_c(m/m_1)$. If the two-body scattering length is negative, a number of states is either zero for $L \ge L_c(m/m_1)$ or infinite for $L<L_c(m/m_1)$. For a finite number of vibrational states, all the binding energies are described by the universal function $\epsilon_{LN}(m/m_1) = {\cal E}(\xi, \eta)$, where $\xi=\displaystyle\frac{N-1/2}{\sqrt{L(L + 1)}}$, $\eta=\displaystyle\sqrt{\frac{m}{m_1 L (L + 1)}}$,and $N$ is the vibrational quantum number. This scaling dependence is in agreement with the numerical calculations for $L > 2$ and only slightly deviates from those for $L = 1, 2$. The universal description implies that the critical values $L_c(m/m_1)$ and $L_b(m/m_1)$ increase as $0.401 \sqrt{m/m_1}$ and $0.563 \sqrt{m/m_1}$, respectively, while a number of vibrational states for $L \ge L_c(m/m_1)$ is within the range $N \le N_{max} \approx 1.1 \sqrt{L(L+1)}+1/2$.