I present the exact solution of a family of fragmented Bose-Hubbard models and represent the models as graphs in one-dimension, two-dimensions and three-dimensions with the condensates in the vertices. The models are solved by the algebraic Bethe ansatz method.
I introduce a new parametrization of a bosonic Lax operator for the algebraic Bethe ansatz method with the gl(2)-invariant R-matrix and use it to present the exact solution of a generalized two-sites Bose-Hubbard model with asymmetric tunnelling. In the no interaction limit I show that the Bethe ansatz equations can be written as a SN−1 sphere, where N is the total number of atoms in the condensate.
The Bose-Hubbard dimer Hamiltonian is a simple yet effective model for describing tunneling phenomena of Bose-Einstein condensates.One of the significant mathematical properties of the model is that it can be exactly solved by Bethe ansatz methods.Here we review the known exact solutions, highlighting the contributions of V.B. Kuznetsov to this field.Two of the exact solutions arise in the context of the Quantum Inverse Scattering Method, while the third solution uses a differential operator realisation of the su(2) Lie algebra.