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

Understand

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.