Exact solution of two bosons in a trap potential: Transition to fragmentation

Fragmentation of a two-boson system in an external potential with contact interaction (i.e., delta-function interaction) is studied using a full variational treatment. The results are compared to those obtained by the standard mean-field method (Gross-Pitaevskii equation) and the more recently developed best mean-field method. The numerically exact calculations show a transition to fragmentation as a function of the boson-boson interaction strength. The Gross-Pitaevskii solution cannot describe fragmentation. The respective energy exhibits a bifurcation which can be viewed as a hint that fragmentation may take place in reality. The best mean-field approach, on the other hand, can predict the fragmentation of the system and give the correct physical description of the fragmented state. The Gross-Pitaevskii energy diverges as the boson-boson interaction goes to infinity, while the exact and the best mean-field energies saturate.