Thin shell Schwarzschild-phantom wormhole stability

It has been suggested that a possible candidate for the present accelerated expansion of the Universe is "phantom energy" . The latter possesses an equation of state of the form $\omega =p/\rho <-1$, consequently violating the null energy condition. As this is the fundamental ingredient to sustain traversable wormholes, this cosmic fluid presents us with a natural scenario for the existence of these exotic geometries. Recently, it has been shown by Lobo that phantom energy with $\omega =p_{r}/\rho <-1$ could support phantom wormholes. Several classes of such solutions have been derived by him. While the inner spacetime is represented by asymptotically flat phantom wormhole that have repulsive gravity, it is most likely to be unstable to perturbations. Hence, we consider a situation, where a phantom wormhole is somehow trapped inside a Schwarzschild sphere across a thin shell. Applying the method developed by Garcia, Lobo and Visser (GLV), we shall exemplify that the shell can possess zones of stability depending on certain constraints.