Analytic properties of the Green's function for the equation of internal gravitational waves in a stratified medium with shear flows

We discuss the problem of constructing the Green's function for the equation for linear internal gravity waves in a finite layer of a stratified medium with background shear flows. We study analytic properties of the Green's function presented as a sum of wave modes (discrete spectrum) and an integral along the cut (continuous spectrum). It is shown that at large times, the contribution from the continuous spectrum of the vertical spectral problem is exponentially small. The convergence of the expansion of the Green's function in wave modes is proved. The asymptotics of the Green's function at large times is shown to be completely determined by the asymptotics of its component wave modes.