Nested algebraic Bethe ansatz for orthogonal and symplectic open spin chains

We present a nested algebraic Bethe ansatz for one-dimensional so _2n- and sp _2n-symmetric open spin chains with diagonal boundary conditions. The monodromy matrix of these spin chains satisfies the defining relations on the extended twisted Yangians X _ρ(so _2n,so _2n ^ρ) ^tw and X _ρ(sp _2n,sp _2n ^ρ) ^tw, respectively. We use a generalisation of the De Vega and Karowski approach allowing us to relate the spectral problem of so _2n- or sp _2n-symmetric open spin chain to that of gl _n-symmetric open spin chain studied by Belliard and Ragoucy. We explicitly derive the structure of Bethe vectors, their eigenvalues and the nested Bethe equations. We also provide a proof of Belliard and Ragoucy's trace formula for Bethe vectors of gl _n-symmetric open spin chains.