  We study the quantum Bethe ansatz equations in the O(2n) sigma-model for
hysical 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} x 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 x 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 N=4 super Yang-Mills theory. We discuss the prospects of
generalization of these Bethe equations to the full superstring sigma-model.
