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

N. N. Konechnaja ; K. A. Mirzoev ; A. A. Shkalikov

Matematičeskie zametki, Tome 104 (2018) no. 2, pp. 231-242

Résumé

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 $$ \begin{aligned} \sigma \in L^1(1,\infty), \qquad \text{if}\quad k , \\ |\sigma|(1+|r|) (1+ |\sigma|) \in L^1(1,\infty), \qquad \text{if}\quad k = n. \end{aligned} $$ 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 $n=1$.

Détail

Keywords: differential operators with distribution coefficients, quasi-derivatives, asymptotics of solutions of differential equations, deficiency index of a differential operator.