Exact Solution Of Two Site Bose-Hubbard Model With General Boundaries

In this paper, we study the two-site Bose-Hubbard model with generic integrable open boundaries, which is a paradigm for the study of strongly correlated bosonic system. The open boundaries of this model are specified by the most general non-diagonal reflecting matrices. Besides the inhomogeneous parameters, the model itself have three free boundary parameters which break the U(1)-symmetry, in other words, break the particle number conservation. Quantum integrable systems without U(1)-symmetry, although they were known to be integrable for many years, the challenge has been to find their general Bethe ansatz solution. The off-diagonal Bethe ansatz (ODBA) methode was proposed in recent years, which can be used to solve the integrale models without U(1)-symmetry. In this paper, the Hamiltonian H under these circumstances is constructed, and we successfully obtained the corresponding Bethe ansatz equations (BAEs) as well as the eigenvalues based on the ODBA method which was generalized to the high spin chain.The paper is organized as follows:In chapter one. firstly, we show a brief introduction of quantum integrable model, Bose-Hubbard model and the recent research status of the spin-s Heisenberg chain with generic non-diagonal boundaries; In chapter two, we firstly give a introduction of the basic symbol of quantum integrable system and some basis properties of transfer matrices; In chapter three, after briefly reviewing the basic ingredients of the ODBA method we give a biref introduction of works on applying the ODBA method to su(2)-invariant spin-s chain; The last chapter, the Hamiltonian H of the two-site Bose-Hubbard model with the generic boundaries was constructed by using the properties of transfer matrices, and base on the ODBA method which was generalized to the high spin chain, we successfully obtained the eigenvalues as well as the corresponding Bethe ansatz equations.