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Using orthogonality of the Legendre polynomials it can be expressed as 

where the rescaled energy ϵ$_{α}$ is given by ϵ$_{α}$ ≡ λE$_{α}$/ (2¯ hω ⊥ ) and λ ≡ ( L/a ⊥ ) 2 is the aspect ratio. For the Fermi-Huang interaction the eigenenergy equa- 

tion (4) attains the form 

In the limit r → 0, the eigenstate is spherically symmet- ric. This allows us to deal with the z -axis only (see [6]). Substitution of the noninteracting wavefunctions (6) and (7) and energies (8) with the subsequent summation over l leads to 

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This sum contains both the regular part and the irregular one, the latter being proportional to z − $^{1}$. The regular part can be extracted using the identity 

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(see [6]) where ζ ( ν, α ) is the Hurwitz zeta function (see, e. g., [7, 8]. Finally we arrive at the transcendental equa- tion for the eigenenergies (4) in [1]. The summands in the sums Eqs. (3) and (4) in [1] and in Eq. (10) decay exponentially with n , leading to the fast converging se- ries. Note that the imaginary parts of the two terms in the left hand side of Eq. (4) in [1] cancel each other automatically. 

[1] V. A. Yurovsky and M. Olshanii, Restricted Thermaliza- tion for Two Interacting Atoms in a Multimode Harmonic Waveguide. 

[2] P. ˇ Seba, Phys. Rev. Lett. 64 , 1855 (1990). 

[3] S. Albeverio and P. Kurasov., Singular perturbations of differential operators : solvable Schr¨ oinger type operators , no. 271 in London Mathematical Society lecture note series (University Press, Cambridge, 2000). 

[4] S. Albeverio and P. ˇ Seba, J. Stat. Phys. 64 , 369 (1991). 

[5] M. Olshanii, Phys. Rev. Lett. 81 , 938 (1998). 

[6] M. G. Moore, T. Bergeman, and M. Olshanii, J. Phys. IV (France) 116 , 69 (2004). 

[7] V. A. Yurovsky, M. Olshanii, and D. S. Weiss, in Adv. At. Mol. Opt. Phys. (Elsvier Academic Press, New York, 2008), vol. 55, pp. 61–138. 

[8] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Se- ries, and Products (Academic Press, New York, 1994). 

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