On Bäcklund transformations and boundary conditions associated with the quantum inverse problem for a discrete nonlinear integrable system and its connection to Baxter’s Q-operator

A discrete nonlinear system is analysed in case of open chain boundary conditions at the ends. It is shown that the integrability of the system remains intact, by obtaining a modified set of Lax equations which automatically take care of the boundary conditions. The same Lax pair also conforms to the conditions stipulated by Sklyanin [5]. The quantum inverse problem is set up and the diagonalisation is carried out by the method of sparation of variables. Bäcklund transformations are then derived under the modified boundary conditions using the classical r-matrix . Finally by quantising the Bäcklund transformation it is possible to identify the relation satisfied by the eigenvalue of Baxter’s Q-operator even for the quasi periodic situation.