We study the quantum Bethe ansatz equations in the $O(2n)$ sigma-model for physical particles on a circle, with the interaction given by the Zamolodchikovs’ S-matrix, in view of its application to quantization of the string on the $S^{2n-1} times R_t$ space. For a finite number of particles, the system looks like an inhomogeneous integrable $O(2n)$ spin chain. Similarly to $OSp(2m+n|2m)$ conformal sigma-model considered by Mann and Polchinski, we reproduce in the limit of large density of particles the finite gap Kazakov-Marshakov-Minahan-Zarembo solution for the classical string and its generalization to the $S^5 times R_t$ sector of the Green-Schwarz-Metsaev-Tseytlin superstring. We also reproduce some quantum effects: the BMN limit and the quantum homogeneous spin chain similar to the one describing the bosonic sector of the one-loop ${cal N}=4$ super Yang-Mills theory. We discuss the prospects of generalization of these Bethe equations to the full superstring sigma-model.

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