The ratio of two zeta-determinants of Dirac Laplacians associated with unitary involutions on a compact manifold with cylindrical end

Given two unitary involutions $\sigma_{1}$ and $\sigma_{2}$ satisfying $G \sigma_{i} = - \sigma_{i} G$ on $ker B$ on a compact manifold with cylindrical end, M. Lesch, K. Wojciechowski ([LW]) and W. M\"uller ([M]) established the formula describing the difference of two eta-invariants with the APS boundary conditions associated with $\sigma_{1}$ and $\sigma_{2}$. In this paper we establish the analogous formula for the zeta-determinants of Dirac Laplacians. For the proof of the result we use the Burghelea-Friedlander-Kappeler's gluing formula for zeta-determinants and the scattering theory developed by W. M\"uller in [M]. This result was also obtained independently by J. Park and K. Wojciechowski ([PW2]).