Exact ground state of finite BoseEinstein condensates on a ring
The stationary Schrödinger equation for N spinless bosons
interacting via pairwise delta function
interaction on a
onedimensional ring is given by:
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
manybody
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 TonksGirardeau 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(N1)/(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 TonksGirardeau 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 LiebLiniger model
[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)
