Abstract.
The discrete energy spectra of composite inverse power-law binding potentials of the form \(V(r;\alpha,\beta,n) =-\alpha/r^{2}+\beta/r^{n}\) with \(n > 2\) are studied analytically. In particular, using a functional matching procedure for the eigenfunctions of the radial Schrödinger equation, we derive a remarkably compact analytical formula for the discrete spectra of binding energies \(\{E(\alpha,\beta, n;k)\}^{k=\infty}_{k=1}\) which characterize the highly excited bound-state resonances of these long-range binding potentials. Our results are of practical importance for the physics of polarized molecules, the physics of composite polymers, and also for physical models describing the quantum interactions of bosonic particles.
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References
H.R. Thorsheim, J. Weiner, P.S. Julienne, Phys. Rev. Lett. 58, 2420 (1987)
J.D. Miller, R.A. Cline, D.J. Heinzen, Phys. Rev. Lett. 71, 2204 (1993)
R.A. Cline, J.D. Miller, D.J. Heinzen, Phys. Rev. Lett. 73, 632 (1994)
C.J. Williams, P.S. Julienne, J. Chem. Phys. 101, 2634 (1994)
R. Cote, A. Dalgarno, M.J. Jamieson, Phys. Rev. A 50, 399 (1994)
H. Wang, P.L. Gould, W.C. Stwalley, Phys. Rev. A 53, R1216 (1996)
B. Gao, Phys. Rev. A 58, 1728 (1998)
J. Trost, C. Eltschka, H. Friedrich, J. Phys. B: At. Mol. Opt. Phys. 31, 361 (1998)
B. Gao, Phys. Rev. A 59, 2778 (1999)
B. Gao, Phys. Rev. Lett. 83, 4225 (1999)
L.D. Landau, E.M. Lifshitz, Quantum Mechanics. Non-relativistic Theory (Mir, Moscow, 1974)
J.E. Lennard-Jones, Proc. R. Soc. London A 106, 463 (1924)
Mott, Massey, Theory of Atomic Collisions (Clarendon Press, Oxford, 1949) p. 30
K.M. Case, Phys. Rev. 80, 797 (1950)
J.-M. Lévy-Leblond, Phys. Rev. 153, 1 (1967)
O.H. Crawford, Proc. Phys. Soc. London 91, 279 (1967)
C. Desfrancois, H. Abdoul-Carime, N. Khelifa, J.P. Schermann, Phys. Rev. Lett. 73, 2436 (1994)
H.E. Camblong, L.N. Epele, H. Fanchiotti, C.A. Garcia Canal, Phys. Rev. Lett. 85, 1590 (2000)
H.E. Camblong, L.N. Epele, H. Fanchiotti, C.A. Garcia Canal, Phys. Rev. Lett. 87, 220402 (2001)
V. Efimov, Phys. Lett. B 33, 563 (1970)
V. Efimov, Nucl. Phys. A 210, 157 (1973)
T. Kraemer et al., Nature 440, 315 (2006)
E. Marinari, G. Parisi, Europhys. Lett. 15, 721 (1991)
C. Nisoli, A.R. Bishop, Phys. Rev. Lett. 112, 070401 (2014)
M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions (Dover Publications, New York, 1970)
Y. Nishida, Y. Kato, C.D. Batista, Nat. Phys. 9, 93 (2013)
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Hod, S. Energy spectrum of the long-range Lennard-Jones potential. Eur. Phys. J. Plus 133, 489 (2018). https://doi.org/10.1140/epjp/i2018-12319-0
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DOI: https://doi.org/10.1140/epjp/i2018-12319-0