Asymptotics of solution to the nonstationary Schrödinger equation

The Cauchy problem with a rapidly oscillating initial condition for the homogeneous Schr?dinger equation was studied in [5]. Continuing the research ideas of this work and [3], in this paper we construct the asymptotic solution to the following mixed problem for the nonstationary Schr?dinger equation: Lhu ? ih?tu + h2?2xu-b(x,t)u = f(x,t), (x,t) ? ??= (0,1) x (0,T], u|t=0 = g(x), u|x=0 = u|x=1 = 0, (1) where h > 0 is a Planck constant, u = u(x,t,h). b(x,t), f(x,t) ? C??(??), g(x) ? C? [0,1] are given functions. The similar problem was studied in [7, 8] when the Plank constant is absent in the first term of the equation and asymptotics of solution of any order with respect to a parameter was constructed. In this paper, we use a generalization of the method used in [7].