Asymptotics of the Solutions of the Sturm--Liouville Equation with Singular Coefficients

Abstract

We obtain asymptotic representations as $\lambda \to \infty$ in the upper and lower half-planes for the solutions of the Sturm--Liouville equation $$-y"+p(x)y^{\prime }+q(x)y={\lambda }^2\rho (x)y,x\in [a,b]\subset \mathbb{R},$$ under the condition that $q$ is a distribution of the first-order singularity, $\rho$ is a positive absolutely continuous function, and $p$ belongs to the space $L_2[a,b]$. In supplementary part, the results are generilized on equation of the following type $$-(r^2{y}^{\prime }{)}^{\prime }+p{y}^{\prime }+qy={\lambda }^2{\rho }^{2}y,x\in [a,b]\subset \mathbb{R},$$ where ${\lambda }^2$ is the large parameter, $r$ and $\rho$ are positive functions, while $p$ and $q$ are complex valued ones. It is assumed that $p\in L_1[a,b],q\in W_2^{-1}[a,b],\rho ,r\in AC[a,b]=W_1^1[a,b],$ moreover, \begin{equation*} ρ'u, r'u, pu \in L_1[a,b], \quad \text{where }\, u=\int q \, dx, \end{equation*} and the antiderivative is understood in the sense of distributions.

11 pages, in Russian + supplement in english