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(Quantum) integrable systems, that belong to solutions to the Yang-Baxter-equation, are often solved by the (algebraic) Bethe Ansatz. Solutions to the Bethe-equations lead to the eigenvalues of the transfermatrix and in that way also to the spectrum of the hamiltonian.

However, suppose you already knew an eigenvalue of the transfermatrix, is there a way to construct the Bethe-roots, that would have lead to it?

If any additional explanations are needed to answer the question, just let me know. I think the answer to the question is in general no. If there are however some specific exactly solvable models, where it is possible, that would be also of interest.

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Yes, there is a method known as McCoy’s method. You can find a detailed description in section 2.2 of the paper http://arxiv.org/abs/hep-th/0307095 . The idea is to use the fact that the eigenvalues of the transfer matrix are (Laurent) polynomials together with the Baxter TQ equation. Knowing a priory the eigenvalues allows you to use the TQ equation to find the zeros of the Q polynomial (which are the Bethe roots); this is easier then solve the Bethe equations directly.

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