Near-extremal and extremal quantum-corrected two-dimensional charged black holes

We consider charged black holes within dilaton gravity with exponential-linear dependence of action coefficients on dilaton and minimal coupling to quantum scalar fields. This includes, in particular, CGHS and RST black holes in the uncharged limit. For non-extremal configuration quantum correction to the total mass, Hawking temperature, electric potential and metric are found explicitly and shown to obey the first generalized law. We also demonstrate that quantum-corrected extremal black holes in these theories do exist and correspond to the classically forbidden region of parameters in the sense that the total mass $M_{tot}<Q$ ($Q$ is a charge). We show that in the limit $T_{H}\to 0$ (where $T_{H}$ is the Hawking temperature) the mass and geometry of non-extremal configuration go smoothly to those of the extremal one, except from the narrow near-horizon region. In the vicinity of the horizon the quantum-corrected geometry (however small quantum the coupling parameter $\kappa$ would be) of a non-extremal configuration tends to not the quantum-corrected extremal one but to the special branch of solutions with the constant dilaton (2D analog of the Bertotti-Robinson metric) instead. Meanwhile, if $\kappa =0$ exactly, the near-extremal configuration tends to the extremal one. We also consider the dilaton theory which corresponds classically to the spherically-symmetrical reduction from 4D case and show that for the quantum-corrected extremal black hole $M_{tot}>Q$.