On the Asymptotic Behavior of Solutions to Two-Term Differential Equations with Singular Coefficients

Asymptotic formulas as $x\to \infty$ are obtained for a fundamental system of solutions to equations of the form
$$\begin{equation*} l(y): = (-1)^n(p(x)y^{(n)})^{(n)}+q(x)y=\lambda y, \qquad x\in [1,\infty), \end{equation*}$$
where $p$ is a locally integrable function representable as
$$p(x) = (1+r(x))^{-1},\qquad r\in L^1(1,\infty),$$
and $q$ is a distribution such that $q= \sigma^{(k)}$ for a fixed integer $k$, $0\leqslant k\leqslant n$, and a function $\sigma$ satisfying the conditions
$$\begin{aligned} \sigma &\in L^1(1,\infty), \qquad \text{if}\quad k <n, \\ |\sigma|(1+|r|) (1+ |\sigma|) &\in L^1(1,\infty), \qquad \text{if}\quad k = n. \end{aligned}$$
Similar results are obtained for functions representable as
$$p(x) = x^{2n+\nu}(1+ r(x))^{-1},\qquad q= \sigma^{(k)},\qquad \sigma(x)=x^{k+\nu} (\beta +s(x)),$$
for fixed $k$, $0\leqslant k\leqslant n$, where the functions $r$ and $s$ satisfy certain integral decay conditions. Theorems on the deficiency index of the minimal symmetric operator generated by the differential expression $l(y)$ (for real functions $p$ and $q$) and theorems on the spectra of the corresponding self-adjoint extensions are also obtained. Complete proofs are given only for the case $n=1$.