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We study the generalized supersymmetric $t-J$ model with boundaries in threedifferent gradings: FFB, BFF and FBF. Starting from the trigonometric R-matrix,and in the framework of the graded quantum inverse scattering method (QISM), wesolve the eigenvalue problems for the supersymmetric $t-J$ model. A detailedcalculations are presented to obtain the eigenvalues and Bethe ansatz equationsof the supersymmetric $t-J$ model with boundaries in three differentbackgrounds.Comment: Latex file, 32 page

physical review. b, condensed matter

Exact diagonalization of the generalized supersymmetric<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>t</mml:mi><mml:mo>−</mml:mo><mml:mi>J</mml:mi></mml:math>model with boundaries

Exact diagonalization of the generalized supersymmetrict−Jmodel with boundaries