Asymptotic Integration of Symmetric Second-Order Quasidifferential Equations

This paper presents conditions on the coefficients of the equations
\begin{align*} -(p(f'-rf))'-\overline{r}p(f'-rf)+qf&=0, \\ -(P(f'-Rf))'-\overline{R}P(f'-Rf)+Qf&=0, \end{align*}
where $1/p$, $1/P$, $q$, $Q$, $r$, $R\in\mathcal{L}^1_{\mathrm{loc}(\mathbb{R}_+)}$, $p$, $P$, $q$, and $Q$ are real-valued functions, while $r$ and $R$ are complex-valued functions, as well as on the fundamental system of solutions of the second equation, which ensure the asymptotic proximity of the solutions of these equations. The results obtained are applied to the study of the spectral properties of the differential operator generated by the expression
$$ -y''+ \sum_{k=0}^{+\infty}h_k\delta(x-x_k)y,\qquad x_k \in \mathbb{R}_+,\quad h_k \in R, $$
in the space $\mathcal{L}^2(\mathbb{R}_+)$. In particular, we obtain conditions on $h_k$, $x_k$ under which the limit-disk case is realized for this operator.