Bethe vectors and recurrence relations for twisted Yangian based models

We study Olshanski twisted Yangian based models, known as one-dimensional “soliton non-preserving” open spin chains, by means of algebraic Bethe ansatz. The even case, when the bulk symmetry is ${\mathfrak{g}\mathfrak{l}}_{2n}$ and the boundary symmetry is ${\mathfrak{s}\mathfrak{p}}_{2n}$ or ${\mathfrak{g}\mathfrak{l}}_{2n}$, was studied in arXiv:1710.08409. In the present work, we focus on the odd case, when the bulk symmetry is ${\mathfrak{g}\mathfrak{l}}_{2n+1}$ and the boundary symmetry is ${\mathfrak{s}\mathfrak{o}}_{2n+1}$. We explicitly construct Bethe vectors and present a more symmetric form of the trace formula. We use the composite model approach and $Y({\mathfrak{g}\mathfrak{l}}_n)$-type recurrence relations to obtain recurrence relations for twisted Yangian based Bethe vectors, for both even and odd cases.