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

We obtain asymptotic formulas for the solutions of the one-dimensional Schrödinger equation − y″ +q(x)y = 0 with oscillating potential q(x)=xβP(x1+α)+cx−2 as x→ +∞. The real parameters α and β satisfy the inequalities β − α ≥ −1, 2α − β > 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.