  Models with an extra dimension generally contain background scalar fields in
a non-trivial configuration, whose stability must be ensured. With gravity
present, the extra dimension is warped by the scalars, and the spin-0 degrees
of freedom in the metric mix with the scalar perturbations. Where possible, we
formally solve the coupled Schrodinger equations for the zero modes of these
spin-0 perturbations. When specialising to the case of two scalars with a
potential generated by a superpotential, we are able to fully solve the system.
We show how these zero modes can be used to construct a solution matrix, whose
eigenvalues tell whether a normalisable zero mode exists, and how many negative
mass modes exist. These facts are crucial in determining stability of the
corresponding background configuration. We provide examples of the general
analysis for domain-wall models of an infinite extra dimension and domain-wall
soft-wall models. For 5D models with two scalars constructed using a
superpotential, we show that a normalisable zero mode survives, even in the
presence of warped gravity. Such models, which are widely used in the
literature, are therefore phenomenologically unacceptable.
