  The Linet-Tian metrics are solutions of the Einstein equations with a
cosmological constant, $\Lambda$, that can be positive or negative. The linear
instability of these metrics in the case $\Lambda <0$, has already been
established. In the case $\Lambda>0$, it was found in a recent analysis that
the perturbation equations admit unstable modes. The analysis was based on the
construction of a gauge invariant function of the metric perturbation
coefficients, called here $W(y)$. This function satisfied a linear second order
equation that could be used to set up a boundary value problem determining the
allowed, real or purely imaginary frequencies for the perturbations.
Nevertheless, the relation of these solutions to the full spectrum of
perturbations, and, therefore, to the evolution of arbitrary perturbations,
remained open. In this paper we consider again the perturbations of the
Linet-Tian metric with $\Lambda >0$, and show, using a form of the Darboux
transformation, that one can associate with the perturbation equations a self
adjoint problem that provides a solution to the completeness and spectrum of
the perturbations. This is also used to construct the explicit relation between
the solutions of the gauge invariant equation for $W(y)$, and the evolution of
arbitrary initial data, thus solving the problem that remained open in the
previous study. Numerical methods are then used to confirm the existence of
unstable modes as a part of the complete spectrum of the perturbations, thus
establishing the linear gravitational instability of the Linet-Tian metrics
with $\Lambda >0$.
