Non-unitary classical r-matrices and Gaudin model with boundary
Following Sklyanin's proposal in the periodic case, our derivation of the generating function of the Gaudin Hamiltonians with boundary terms, both in the rational and trigonometric case, is centered on the quasi-classical expansion of the linear combination of the transfer matrix of the XXX/XXZ Heisenberg spin chain and the corresponding central element, the so-called Sklyanin determinant. The linear bracket for the modified Lax matrix can be defined using the corresponding non-unitary classical r-matrix. This bracket is by definition anti-symmetric and it also obeys the Jacobi identity. Therefore, the entries of the modified Lax matrix generate an infinite dimensional Lie algebra, the generalized Gaudin algebra. Using suitable a set of generators of this algebra, in the appropriate realization, we define the Bethe vectors which yield strikingly simple off shell action of the generating function. Therefore, we fully implement the algebraic Bethe ansatz, obtaining the spectrum of the generating function and the corresponding Bethe equations.