JETP Letters: issues online

Universal description of the rotational-vibrational spectrum of three particles with zero-range interactions
O. I. Kartavtsev, A. V. Malykh Joint Institute for Nuclear Research, 141980 Dubna, Russia
PACS: 03.65.Ge, 03.75.Ss, 21.45.+v, 36.90.+f
Abstract A comprehensive universal description of the rotational-vibrational spectrum for two identical particles of mass m and the third particle of mass m₁ in the zero-range limit of the interaction between different particles is given for arbitrary values of the mass ratio m/m₁ and the total angular momentum L. It is found that the number of vibrational states is determined by the functions L_c(m/m₁) and L_b(m/m₁). Explicitly, if the two-body scattering length is positive, the number of states is finite for $L_c(m/m_1) \le L \le L_b(m/m_1)$ , zero for L > L_b(m/m₁), and infinite for L < L_c(m/m₁). If the two-body scattering length is negative, the number of states is zero for $L \ge L_c(m/m_1)$ and infinite for L < L_c(m/m₁). For the finite number of vibrational states, all the binding energies are described by the universal function $\varepsilon_{L N}(m/m_1) = {\cal E}(\xi, \eta)$ , where $\xi = {(N - 1/2)}/{\sqrt{L(L + 1)}}$ , $\eta =\sqrt{{m}/{m_1 L (L + 1)}}$ , 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/m₁) and L_b(m/m₁) increase as $0.401 \sqrt{m/m_1}$ and $0.563 \sqrt{m/m_1}$ , respectively, while the number of vibrational states for $L \ge L_c(m/m_1)$ is within the range $N \le N_{\max} \approx 1.1 \sqrt{L(L + 1)} + 1/2$ .

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