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

O. I. Kartavtsev; A. V. Malykh

doi:10.1134/S002136400722002X

paper

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

arXiv:0709.4151 · doi:10.1134/S002136400722002X

Abstract

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 $L_{c} (m / m_{1}) \leq L \leq L_{b} (m / m_{1})$ , 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 $L \geq L_{c} (m / m_{1})$ 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 $ε_{L N} (m / m_{1}) = E (ξ, η)$ , where $ξ = \frac{N - 1/2}{L ( L + 1 )}$ , $η = \frac{m}{m _{1} L ( L + 1 )}$ ,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 $0.401 m / m_{1}$ and $0.563 m / m_{1}$ , respectively, while a number of vibrational states for $L \geq L_{c} (m / m_{1})$ is within the range $N \leq N_{ma x} \approx 1.1 L (L + 1) + 1/2$ .

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

Abstract

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