Asymptotics of Solutions of Two-Term Differential Equations

Abstract

Asymptotic formulas for the fundamental solution system as $x\to\infty$ are obtained for equations of the form

$$l(y):=(-1)^n(p(x)y^{(n)})^{(n)}+q(x)y=\lambda y,\qquad x\in[1,\infty),$$

where $p$ is a locally integrable function admitting the representation

$$p(x)=(1+r(x))^{-1},\qquad r\in L^1 [1,\infty),$$

and $q$ is a distribution representable for some given $k$, $0\le k\le n$, as $q=\sigma^{(k)}$, where

$$\begin{alignedat}{2} \sigma&\in L^1[1,\infty)&\qquad &\text{if }k<n, \\ |\sigma|(1+|r|)(1+|\sigma|)&\in L^1[1,\,\infty) &\qquad &\text{if }k=n. \end{alignedat}$$

Similar results are obtained for the equations $l(y)=\lambda y$ whose coefficients $p(x)$ and $q(x)$ admit the following representation for a given $k$, $0\le k\le n$:

$$p(x)=x^{2n+\nu}(1+r(x))^{-1},\qquad q=\sigma^{(k)},\quad \sigma(x)=x^{k+\nu}(\beta+s(x)),$$

where the functions $r$ and $s$ satisfy certain integral decay conditions. We also obtain theorems on the deficiency indices of the minimal symmetric operator generated by the differential expression $l(y)$ (with real functions $p$ and $q$) as well as theorems on the spectra of the corresponding self-adjoint extensions.