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

Abstract

Asymptotic formulas as x→∞ are obtained for a fundamental system of solutions to equations of the form

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

, where p is a locally integrable function representable as

\(p\left( x \right) = {\left( {1 + r\left( x \right)} \right)^{ - 1}},r \in {L^1}\left( {1,\infty } \right)\)

, and q is a distribution such that q = σ^(k) for a fixed integer k, 0 ≤ k ≤ n, and a function σ satisfying the conditions \(\sigma \in {L^1}\left( {1,\infty } \right)ifk < n,\)\(\left| \sigma \right|\left( {1 + \left| r \right|} \right)\left( {1 + \left| \sigma \right|} \right) \in {L^1}\left( {1,\infty } \right)ifk = n\). Similar results are obtained for functions representable as

\(p\left( x \right) = {x^{2n + v}}{\left( {1 + r\left( x \right)} \right)^{ - 1}},q = {\sigma ^{\left( k \right)}},\sigma \left( x \right) = {x^{k + v}}\left( {\beta + s\left( x \right)} \right)\)

, for fixed k, 0 ≤ k ≤ 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.