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

Abstract

A comprehensive universal description of the rotational-vibrational spectrum for two identical particles of mass m and a third particle of 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. It is found that the number of vibrational states is determined by the functions L c(m/m 1) and L b(m/m 1). Explicitly, if the two-body scattering length is positive, the number of states is finite for L c(m/m 1) ≤ L ≤ 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, the number of states is zero for L ≥ L c(m/m 1) and infinite for L < L c(m/m 1). For the finite number of vibrational states, all the binding energies are described by the universal function ɛLN(m/m 1) = (ξ, η), where ξ = (N − 1/2)/√L(L + 1), η = √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 the number of vibrational states for L ≥ L c(m/m 1) is within the range N ≤ N max ≈ 1.1√L(L + 1) + 1/2.