Nested algebraic Bethe ansatz for open spin chains with even twisted Yangian symmetry

We present a nested algebraic Bethe ansatz for a one dimensional open spin chain whose boundary quantum spaces are irreducible $\mathfrak{so}_{2n}$- or $\mathfrak{sp}_{2n}$-representations and the monodromy matrix satisfies the defining relations of the Olshanskii twisted Yangian $Y^\pm(\mathfrak{gl}_{2n})$. We use a generalization of the Bethe ansatz introduced by De Vega and Karowski which allows us to relate the spectral problem of a $\mathfrak{so}_{2n}$- or $\mathfrak{sp}_{2n}$-symmetric open spin chain to that of a $\mathfrak{gl}_{n}$-symmetric periodic spin chain. We explicitly derive the structure of the Bethe vectors and the nested Bethe equations.