  The spectrum of the light-cone AdS_5 \times S^5 superstring contains states
composed of particles with complex momenta including in particular those which
turn into bound states in the decompactification limit. We propose the mirror
TBA description for these states. We focus on a three-particle state which is a
finite-size representative of a scattering state of a fundamental particle and
a two-particle bound state and dual to an operator from the su(2) sector of N=4
SYM. We find that the analytic behavior of Y-functions differs drastically from
the case of states with real momenta. Most importantly, Y_Q-functions exhibit
poles in the analyticity strip which leads to the appearance of new terms in
the formula for the energy of this state. In addition, the TBA equations are
supplied by quantization conditions which involve Y_2. Considering yet another
example of a three-particle state, we find that the corresponding quantization
conditions do not even involve Y_1. Our treatment can be generalized to a wide
class of states with complex momenta.
