Asymptotic integration of the second order differential equation, resonance effect

Abstract

There will be presented asymptotic formulas for solutions of the equation
$$

y^'' + ( 1 + \varphi (x) ) y = 0,   0  <   x_0  <  x  <  \infty ,

$$
where function $\varphi$ is small in a certain sense for large values of the argument. Usage of method of L-diagonal systems allows to obtain various forms of solutions depending on the properties of function $\varphi$.
The main aim will be discussion about the second order differential equations possesing a resonance effect known for Wigner-von Neumann potential. A class of potentials generalizing that of Wigner-von Neumann will be presented.