2 = − 1 √$_{λ}$ ζ ( 1 2 , − ϵ λ ) (13) C$_{α}$ = 2 a $_{⊥}$√$_{πL}$ [ ∞ ∑ n =0 ( $_{cot}$√$_{ϵ}$$_{α}$ − λn ( ϵ$_{α}$ − λn ) 3 / 2 + 1 ( ϵ$_{α}$ − λn ) sin 2 √$_{ϵ}$$_{α}$ − λn )] − 1 / 2 , (10) ∂ ∂r   r ∑ n,l 〈 ρ | n 〉〈 z | l 〉〈 n | 0 〉〈 0 | 0 〉 E − E$_{nl}$   r =0 = 1 V (11) √$_{λ}$ ∂ ∂z [ z ∞ ∑ n =0 cos ( $_{2}$√$_{ϵ}$$_{α}$ − λn ( z/L − 1 / 2) ) √$_{ϵ}$$_{α}$ − λn $_{sin}$√$_{ϵ}$$_{α}$ λn ]$_{z}$$_{=0}$ = a$_{s}$ a ⊥ . (12) lim z → 0 [ ∞ ∑ n =0 ( λn − ϵ ) − 1 / $^{2}$exp( − $_{2}$√$_{λn}$ − ϵz/L ) − L λz ] Using orthogonality of the Legendre polynomials it can be expressed as (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. 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 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 [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).