  The stability of certain class of asymptotically AdS wormholes in vacuum
against scalar field perturbations is analyzed. For a free massive scalar
field, the stability of the perturbation is guaranteed provided the squared
mass is bounded from below by a negative quantity. Depending on the base
manifold of the AdS asymptotics, this lower bound could be more stringent than
the Breitenlohner-Freedman bound. An exact expression for the spectrum is found
analytically. For a scalar field perturbation with a nonminimal coupling, slow
fall-off asymptotic behavior is also allowed, provided the squared mass
fulfills certain negative upper bound. Although the Ricci scalar is not
constant, an exact expression for the spectrum of the scalar field can also be
found, and three different quantizations for the scalar field can be carried
out. They are characterized by the fall-off of the scalar field, which can be
fast or slow with respect to each asymptotic region. For these perturbations,
stability can be achieved in a range of negative squared masses which depends
on the base manifold of the AdS asymptotics. This analysis also extends to a
class of gravitational solitons with a single conformal boundary.
