How coordinate Bethe ansatz works for Inozemtsev model

Three decades ago, Inozemtsev found an isotropic long-range spin chain with elliptic pair potential that interpolates between the Heisenberg and Haldane-Shastry (HS) spin chains while admitting an exact solution throughout, based on a connection with the elliptic quantum Calogero-Sutherland model. Though Inozemtsev's spin chain is widely believed to be quantum integrable, the underlying algebraic reason for its exact solvability is not yet well understood. As a step in this direction we refine Inozemtsev's `extended coordinate Bethe ansatz' and clarify various aspects of the model's exact spectrum and its limits. We identify quasimomenta in terms of which the $M$-particle energy is close to being (functionally) additive, as one would expect from the limiting models; our expression is additive iff the energy of the elliptic Calogero-Sutherland system is so. This enables us to rewrite the energy and Bethe-ansatz equations on the elliptic curve, turning the spectral problem into a rational problem as might be expected for an isotropic spin chain. We treat the $M=2$ particle sector and its limits in detail. We identify an $S$-matrix that is independent of positions. We show that the Bethe-ansatz equations reduce to those of Heisenberg in one limit and give rise to the `motifs' of HS in the other limit. We show that, as the interpolation parameter changes, the `scattering states' from Heisenberg become Yangian highest-weight states for HS, while bound states become ($\mathfrak{sl}_2$-highest weight versions of) affine descendants of the magnons from $M=1$. For bound states we find a generalisation of the known equation for the `critical length' for the Heisenberg spin chain. We discuss completeness for $M=2$ by passing to the elliptic curve. Our review of the two-particle sectors of the Heisenberg and HS spin chains may be of independent interest.