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 ${\mathfrak{s}\mathfrak{o}}_{2n}$- or ${\mathfrak{s}\mathfrak{p}}_{2n}$-representations and the monodromy matrix satisfies the defining relations of the Olshanskii twisted Yangian $Y^{\pm }({\mathfrak{g}\mathfrak{l}}_{2n})$. We use a generalization of the Bethe ansatz introduced by De Vega and Karowski which allows us to relate the spectral problem of a ${\mathfrak{s}\mathfrak{o}}_{2n}$- or ${\mathfrak{s}\mathfrak{p}}_{2n}$-symmetric open spin chain to that of a ${\mathfrak{g}\mathfrak{l}}_n$-symmetric periodic spin chain. We explicitly derive the structure of the Bethe vectors and the nested Bethe equations.