Regular on-shell Bethe data without a completeness assumption

For the ordered monodromy on $V_N$ with $V={\mathbb{C}}^2$, define the shifted site operator $L_{aj}(u):=R_{aj}(u-i\mathrm{/}2)$, shifted monodromy $T_a^L(u):=L_{aN}(u)\cdots L_{a1}(u)$, and shifted transfer matrix $\tau (u):={\mathrm{tr}}_a{T}_a^L(u)$. In the ordered auxiliary basis $e_{\uparrow },e_{\downarrow }$, write $T_a^L(u)={\left(\begin{array}{cc}A(u)& B(u)\\ C(u)& D(u)\end{array}\right)}_a,\mathrm{\Omega }:=e_{\uparrow }^{\otimes N}\mathrm{.}$ For an integer $M$ with $0\le M\le N$, a regular on-shell Bethe datum is a tuple of finite complex numbers ${\lambda }_1,\dots ,{\lambda }_M$ satisfying all of the following conditions:

1. The roots are pairwise distinct, none equals $i\mathrm{/}2$ or $-i\mathrm{/}2$, and no difference ${\lambda }_j-{\lambda }_k$ for $j\mathrm{\ne }k$ equals $i$ or $-i$.
2. For each $j=1,\dots ,M$, ${\left(\frac{{\lambda }_j+i\mathrm{/}2}{{\lambda }_j-i\mathrm{/}2}\right)}^N={\prod }_{k\mathrm{\ne }j}\frac{{\lambda }_j-{\lambda }_k+i}{{\lambda }_j-{\lambda }_k-i}\mathrm{.}$
3. The associated vector $\mathrm{\Psi }({\lambda }_1,\dots ,{\lambda }_M):=B({\lambda }_1)\cdots B({\lambda }_M)\mathrm{\Omega }$ is nonzero.

For $M=0$ the root tuple and products are empty and $\mathrm{\Psi }=\mathrm{\Omega }$. This definition records regular solutions and nonzero vectors only; it makes no claim of completeness, covers no singular-root regularization, and does not include descendants generated from other reference vectors.

Ordered auxiliary-space monodromy

Faddeev, §4, equations (66)–(97)

Classical and Quantum Integrable Systems, Lecture 15, §§15.3–15.5