On B\"{a}cklund 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. B\"{a}cklund transformations are then derived under the modified boundary conditions using the classical r-matrix . Finally by quantising the B\"{a}cklund transformation it is possible to identify the relation satisfied by the eigenvalue of Baxter's Q-operator even for the quasi periodic situation.