  The hybrid and the dressed metric formalisms for the study of primordial
perturbations in Loop Quantum Cosmology lead to dynamical equations for the
modes of these perturbations that are of a generalized harmonic-oscillator
type, with a mass that depends on the background but is the same for all modes.
For quantum background states that are peaked on trajectories of the effective
description of Loop Quantum Cosmology, the main difference between the two
considered formalisms is found in the expression of this mass. The value of the
mass at the bounce is especially important, since it is only in a short
interval around this event that the quantum geometry effects on the
perturbations are relevant. In a previous article, the properties of this mass
were discussed for an inflaton potential of quadratic form, or with similar
characteristics. In the present work, we extend this study to other interesting
potentials in cosmology, namely the Starobinsky and the exponential potentials.
We prove that there exists a finite interval of values of the potential (which
includes the zero but typically goes beyond the sector of kinetically dominated
inflaton energy density) for which the hybrid mass is positive at the bounce
whereas the dressed metric mass is negative.
