< phi² > for a scalar field in 2D black holes: a new uniform approximation

We study nonconformal quantum scalar fields and averages of their local observables (such as <phi^2>^{ren} and T_{ab}^{ren}) in a spacetime of a 2-dimensional black hole. In order to get an analytical approximation for these expressions the WKB approximation is often used. We demonstrate that at the horizon WKB approximation is violated for a nonconformal field, that is when the field mass or/and the parameter of non-minimal coupling do not vanish. We propose a new "uniform approximation" which solves this problem. We use this approximation to obtain an improved analytical approximation for <phi^2>^{ren} in the 2-dimensional black hole geometry. We compare the obtained results with numerical calculations.