Dynamics of S=1 antiferromagnetic bond-alternating chains in the dimer phase, in the vicinity of the critical point with the Haldane phase, is studied by a field theoretical method. This model is considered to represent the compound Ni(C$_9$H$_{24}$N$_4$)(NO$_2$)ClO$_4$ (abbreviated as NTENP). We construct the sine-Gordon (SG) field theory as a low-energy effective model of this system, starting from a Tomonaga-Luttinger liquid at the critical point. Using the exact solution of the SG theory, we give a field theoretical picture of the low-energy excitation spectrum of NTENP. Results derived from our picture are in a good agreement with results of inelastic neutron scattering experiments on NTENP and numerical calculation of the dynamical structure factor. Furthermore, on the basis of the obtained theoretical picture, we predict that the sharp peaks correspond to a single elementary excitation are absent in the Raman scattering spectrum of NTENP in contrast to the inelastic neutron scattering spectrum.
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