On Bethe strings in the two-particle sector of the closed SU(2)_q invariant spin chain

In this paper we investigate complex solutions of the Bethe equations in the two-particle sector, both for arbitrary finite number of sites and for the thermodynamic limit. We find the number of complex solutions (strings) and compare it with the string conjecture prediction. Some simple properties of these solutions like position in the spectrum, crossing of levels, connection to the ground state and transformation to the real solutions are discussed. Counting both real and complex solutions, we find expected number of highest-weight Bethe states