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.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
C. Ospelkaus, S. Ospelkaus, K. Sengstock, et al., Phys. Rev. Lett. 96, 020401 (2006).
T. Karpiuk, M. Brewczyk, M. Gajda, et al., J. Phys. B 38, L215 (2005).
Y. Shin, M. W. Zwierlein, C. H. Schunck, et al., Phys. Rev. Lett. 97, 030401 (2006).
F. Chevy, Phys. Rev. Lett. 96, 130401 (2006).
M. Iskin and C. A. R. Sa de Melo, Phys. Rev. Lett. 97, 100404 (2006).
F. M. Cucchietti and E. Timmermans, Phys. Rev. Lett. 96, 210401 (2006).
R. M. Kalas and D. Blume, Phys. Rev. A 73, 043608 (2006).
F. M. Pen’kov, Phys. Rev. A 60, 3756 (1999).
A. S. Jensen and D. V. Fedorov, Europhys. Lett. 62, 336 (2003).
V. Efimov, Nucl. Phys. A 210, 157 (1973).
Yu. N. Ovchinnikov and I. M. Sigal, Ann. Phys. (N.Y.) 123, 274 (1979).
Yong Li, Qingdong Gou, and Tingyun Shi, Phys. Rev. A 74, 032502 (2006).
J. P. D’Incao and B. D. Esry, Phys. Rev. A 73, 030702(R) (2006).
M. Kh. Shermatov, Teor. Mat. Fiz. 136(2), 257 (2003) [Theor. Math. Phys. 136, 1119 (2003)].
J. H. Macek and J. Sternberg, Phys. Rev. Lett. 97, 023201 (2006).
D. S. Petrov, Phys. Rev. A 67, 010703(R) (2003).
D. S. Petrov, C. Salomon, and G. V. Shlyapnikov, Phys. Rev. A 71, 012708 (2005).
O. I. Kartavtsev and A. V. Malykh, J. Phys. B 40, 1429 (2007).
O. I. Kartavtsev and J. H. Macek, Few-Body Syst. 31, 249 (2002).
J. H. Macek, J. Phys. B 1, 831 (1968).
O. I. Kartavtsev, Few-Body Syst. Suppl. 10, 199 (1999).
O. I. Kartavtsev and A. V. Malykh, Phys. Rev. A 74, 042506 (2006).
Yu. N. Demkov and V. N. Ostrovskii, Zero-Range Potentials and Their Applications in Atomic Physics (Leningr. Gos. Univ., Leningrad, 1975; Plenum, New York, 1988).
K. Wódkiewicz, Phys. Rev. A 43, 68 (1991).
Z. Idziaszek and T. Calarco, Phys. Rev. Lett. 96, 013201 (2006).
E. Braaten and H.-W. Hammer, Phys. Rev. A 67, 042706 (2003).
Higher Transcendental Functions (Bateman Manuscript Project), Ed. by A. Erdelyi (McGraw-Hill, New York, 1953; Nauka, Moscow, 1965).
D. Blume, Phys. Rev. B 72, 094510 (2005).
Author information
Authors and Affiliations
Additional information
The text was submitted by the authors in English.
Rights and permissions
About this article
Cite this article
Kartavtsev, O.I., Malykh, A.V. Universal description of the rotational-vibrational spectrum of three particles with zero-range interactions. Jetp Lett. 86, 625–629 (2008). https://doi.org/10.1134/S002136400722002X
Received:
Issue Date:
DOI: https://doi.org/10.1134/S002136400722002X