We study the linear stability of a vertical interface separating two fluid columns of different densities under the influence of gravity. This configuration was investigated recently at Caltech by Gat et al. (2017), who performed direct numerical simulation (DNS) of this flow. Here, we are interested in a theoretical understanding of the early phases of the evolution, where a small-amplitude perturbation analysis can be carried out. In the first limit, we consider immiscible flow with surface tension and obtain a closed-form solution of the time evolution of the interface amplitude. The solution is non-modal and expressible as a function of the parabolic cylinder function. Secondly, we consider the viscous (miscible) case. Initially, we assume quasi-steady state (QSSA) of the base flow and pose the problem as an eigenvalue problem. Subsequently, we carry out adjoint-based optimization of the most amplified eigenmode. This results in an initial condition that leads to the maximum growth of disturbances at a finite time. Preliminary results indicate that the perturbation energy of wave modes with small wave numbers may experience substantial transient growth prior to decaying asymptotically in time, despite the infinitesimal assumption of the linearized problem. It is also found that the maximum growth rate is about one order of magnitude higher than that of the non-optimized case. The sensitivity of perturbation growth with respect to initial time, density, and viscosity ratios will be investigated.