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We have thus shown that ${\cal H}_{\vec{n}}(\vec{p},\vec{\tau})$ is the only one-particle $L^2$ bound state of ${\cal H}_\text{dyn}$ with angular momentum $\vec{n}$ for each $\vec{n}\in\mathbb{Z}^{2k}$. This proves the statement about the existence of a (non-degenerate) eigenvalue for each $\vec{n}\in\mathbb{Z}^{2k}$.
The proof of the statement about the multiplicity is straightforward: Given a pair of $\vec{p}\in \Pi_{1,k}$ and $\vec{\tau}\in\Pi_{2,k}$ such that ${\cal H}_{\vec{n}}(\vec{p},\vec{\tau})$ is non-degenerate, one finds that $\vec{\tau}=(\tau_1,\dots,\tau_{2k})$ satisfies the Bethe ansatz equations (\[BA-1\], \[BA-2\]) for some $\vec{u}=(u_1,\dots,u_{2k})\in\mathbb{C}^{2k}$. The Bethe ansatz equations for this set of the $u_j$’s is consistent with the assumption that ${\cal H}_{\vec{n}}(\vec{p},\vec{\tau})$ is non-degenerate. This yields a set of Bethe ansatz equations for all $\vec{u}\in\mathbb{C}^{ 0b46394aab
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