Construction of the asymptotics of the solutions of the one-dimensional Schrodinger equation with rapidly oscillating potential

We obtain asymptotic formulas for the solutions of the one-dimensional Schrodinger equation -y" + q(x)y = 0 with oscillating potential q(x) = x(beta)P(x(1+alpha),) + cx(-2) as x -> +infinity. The real parameters alpha and beta satisfy the inequalities beta - alpha > -1, 2 alpha - beta > 0 and c is an arbitrary real constant. The real function P(x) is either periodic with period T, or a trigonometric polynomial. To construct the asymptotics, we apply the ideas of the averaging method and use Levinson's fundamental theorem.