On a new approach for studying asymptotic behavior of solutions to singular differential equations

Abstract

In the work we propose a new approach for studying the asymptotic behavior for large $x$ of the solutions to singular linear two-terms differential equations $$ -\frac{d^n}{dx^n}y(x,\lambda)+\lambda q(x)y(x,\lambda)=0 $$ with a potential $q(x)$ non-regular growing as $x\to\infty$. The idea of constructing the asymptotics for the solutions of singular linear differential equations and its effectiveness is demonstrated for 4th order equations with an oscillating potential.