  We develop a theoretical background to treat exciton states of semiconductor
single-walled carbon nanotubes (SWCNTs) in presence of a periodic potential
induced by the surface acoustic wave (SAW) propagating along semiconducting
SWCNT. The formalism naturally accounts for the electronic bands splitting into
the Floquet sub-bands brought about by the Bragg scattering on the SAW
potential. Optically induced transitions within the Floquet states and
formation of correlated electron-hole pairs, i.e., exciton states, are examined
numerically. We discuss dynamical formation of new van Hove singularities
within electron-hole continuum and associated reduction of the exciton
oscillator strengths and its effect on the photoluminescence quenching in
presence of the SAW. We argue that SAW induced dynamical gaps in the single
particle dispersion leads to redistribution of the oscillator strength from
excitons to the Floquet edge states. The simulations also confirm exciton
energy Stark red shift as well as reduction in the binding energy. Comparison
of our results with previous theoretical and experimental studies is provided.
