Quantum focusing conjecture in two-dimensional evaporating black holes

We consider the quantum focusing conjecture (QFC) for two-dimensional evaporating black holes in the Russo-Susskind-Thorlacius (RST) model. The QFC is closely related to the behavior of the generalized entropy. In the context of the black hole evaporation, the entanglement entropy of the Hawking radiation is decreasing after the Page time, and therefore it is not obvious whether the QFC holds. One of the present authors previously addressed this problem in a four-dimensional spherically symmetric dynamical black hole model and showed that the QFC is satisfied. However, the background spacetime considered was approximated by the Vaidya metric, and quantum effects of matters in the semiclassical regime were not fully taken into consideration. It remains to be seen if the QFC in fact holds for exact solutions of the semiclassical Einstein equations. In this paper, we address this problem in the RST model, which allows us to solve the semiclassical equations of motion exactly. We prove that the QFC is satisfied for evaporating black holes in the RST model with the island formation taken into account.