Asymptotics of Solution to the Nonstationary Schrodinger Equation

The Cauchy problem with a rapidly oscillating initial condition for the homogeneous Schrodinger 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 Schrodinger equation:



L(h)u ih partial derivative(t)u + h(2)partial derivative(2)(x)u - b(x,t)u = f(x,t) (x,t) is an element of Omega = (0,1) x (0,t],



u vertical bar(t=0) = g(x), u vertical bar(x=0) = u vertical bar(x=1) = 0



where h > 0 is a Planck constant, u = u(x,t,h). b(x,t), f (x,t) is an element of C-infinity(Omega),g (x) is an element of C-infinity[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].