We study the generalized supersymmetric $t-J$ model with boundaries in three different gradings: fermionic, fermionic, bosonic (FFB), bosonic, fermionic, fermionic (BFF), and fermionic, bosonic, fermionic (FBF). Starting from the trigonometric R matrix, and in the framework of the graded quantum inverse scattering method, we solve the eigenvalue problems for the supersymmetric $t-J$ model. Detailed calculations are presented to obtain the eigenvalues and Bethe ansatz equations of the supersymmetric $t-J$ model with boundaries in three different backgrounds.

• Received 9 June 1999

DOI:https://doi.org/10.1103/PhysRevB.61.3450