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 $-y''+q(x)y=\nobreak 0$ with oscillating potential $q(x)=x^\beta P(x^{1+\alpha})+cx^{-2}$ as $x\to+\nobreak \infty$. The real parameters $\alpha$ and $\beta$ satisfy the inequalities $\beta-\alpha\ge\nobreak -1$, $2\alpha-\beta>\nobreak 0$ 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.