A note on linearized stability of Schwarzschild thin-shell wormholes with variable equations of state

We discuss how the assumption of variable equation of state (EoS) allows the elimination of the instability at equilibrium throat radius $a_0=3M$ featured by previous Schwarzschild thin-shell wormhole models. Unobstructed stability regions are found for three choices of variable EoS. Two of these EoS entail linear stability at every equilibrium radius. Particularly, the thin-shell remains stable as $a_0$ approaches the Schwarzschild radius $2M$. A perturbative analysis of the wormhole equation of motion is carried out in the case of variable Chaplygin EoS. The squared proper angular frequency $\omega_0^2$ of small throat oscillations is linked with the second derivative of the thin-shell potential. In various situations $\omega_0^2$ remains positive and bounded in the limit $a_0\rightarrow 2M$.