  We introduce a microscopic computation which shows that the Hamiltonian of a
Bose-Einstein Condensate can be analytically solved in the two-mode
approximation, in particular, in the case of an asymmetric double-well
condensate in the dilute regime. Our model is exactly diagonalisable when the
overlap of the quasilocalized modes in each well is small enough with respect
to the trap asymmetry. For larger overlaps or highly symmetric traps, our
diagonalisable Hamiltonian acquires extra terms that we treat within
perturbation theory.
