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 $$ -y"+p(x)y'+q(x)y= \lambda ^2 \rho(x)y, \qquad x\in [a,b] \subset \mathbb{R}, $$ 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 $$-(r^2y')'+py'+qy=\lambda ^2 \rho^2 y, \qquad x\in [a,b] \subset \mathbb{R},$$ 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 $p\in L_1[a,b],\quad q\in W_2^{-1}[a,b], \quad \rho ,r \in AC[a,b] =W_1^1[a,b],$ moreover, \begin{equation*} \rho'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.