Note on the Schwarzschild-phantom wormhole

Recently, it has been shown by Lobo, Parsaei and Riazi (LPR) that phantom energy with $\omega =p_{r}/\rho <-1$ could support phantom wormholes. Several classes of such solutions have been derived by them. 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. It turns out that zones corresponding to "force" constraint are more restrictive than those from the "mass" constraint. We shall also enumerate the interior energy content by using the gravitational energy integral proposed by Lynden-Bell, Katz and Bi% \v{c}\'ak. It turns out that, even though the interior mass is positive, the integral implies repulsive energy. This is consistent with the phantom nature of interior matter.