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Exact ground state of finite Bose-Einstein condensates on a ring

The stationary Schrödinger equation for N spinless bosons interacting via pairwise delta function
interaction on a one-dimensional ring is given by:   

                                                     sgl 

where the particle coordinates x_i run from -L/2 to +L/2 and periodic boundary conditions are imposed on the
wave function. Even with this simple interaction potential the many-body Schrödinger equation defies
any accurate description by direct diagonalization techniques except for very weak interaction, see [1].
In 1963 Lieb and Liniger obtained a coupled set of coupled transcendental equations which - when solved numerically - provide an exact solution of the problem [2]. Lieb and Liniger solved these equations for two particles and derived (and solved) an integral equation for the thermodynamic limit of the system. [2]. Since then most of the work on the model has focused on solutions in the thermodynamic limit. For three particles Muga and Snider provided solutions for attractive and repulsive interactions [5]. In our work we solved the problem for the ground state of up to fifty bosons for attractive and repulsive interactions [6]. In the same work we also show (i) how the solution for a finite number of particles relates to the solution in the thermodynamic limit, (ii) how the Tonks-Girardeau limit [4] is obtained when the interaction strength becomes strongly repulsive and (iii) for attractive interactions how the the system relates to the solution of N bosons on an infinte line [3].

In all files the first column contains G=2c(N-1)/(2Pi).
For attractive interaction the second column contains the corresponding energy per particle of N bosons on an infinite  line, see [3] and references therein, and for repulsive interaction the corresponding energy per particle of N bosons in the Tonks-Girardeau limit [4].
The third column contains the exact ground state energy per particle of N bosons on a ring of length 2 Pi.

The ground state energies for rings of any other length can be obtained by using the relation
         

                                                                         

The exact ground state energies for more than three particles can be found in the files below. Please cite Ref. [6]
if the data provided here is helpful in your own work. Read also Elliott Lieb's Scholarpedia article on the Lieb-Liniger model

Particle number
N=2
N=3
N=4
N=5
N=11
N=15
N=25
N=50
Attractive interaction
E/N
E/N
E/N E/N E/N E/N E/N E/N
Repulsive interaction
E/N E/N E/N E/N E/N E/N E/N E/N



[1]  O. E. Alon, A. I. Streltsov, K. Sakmann and L. S. Cederbaum, Europhys. Lett. 67 (2004), 8

[2]  E. H. Lieb and W. Liniger, Phys. Rev. 130, 1605 (1963)

[3]  J. B. McGuire, J. Math. Phys. 5, 622 (1964)
See also the references in F. Calogero, A. Degasperis, Phys. Rev. A 11, 265 (1975)                                                       

[4]  M. Girardeau, J. Math. Phys. 1, 516 (1960)                                                    

[5]  J. G. Muga and R. F. Snider, Phys. Rev. A 57, 3317 (1998)

[6]  K. Sakmann , A. I. Streltsov, O. E. Alon and L. S. Cederbaum, Phys. Rev. A 72, 033613 (2005)




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