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

Asymptotic formulas as x →∞ are obtained for a fundamental system of solutions to equations of the form l ( y ) : = ( − 1 ) n ( p ( x ) y ( n ) ) ( n ) + q ( x ) y = λ y , x ∈ [ 1 , ∞ ) , where p is a locally integrable function representable as p ( x ) = ( 1 + r ( x ) ) − 1 , r ∈ L 1 ( 1 , ∞ ) , 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 σ ∈ L 1 ( 1 , ∞ ) i f k < n , | σ | ( 1 + | r | ) ( 1 + | σ | ) ∈ L 1 ( 1 , ∞ ) i f k = n . Similar results are obtained for functions representable as p ( x ) = x 2 n + v ( 1 + r ( x ) ) − 1 , q = σ ( k ) , σ ( x ) = x k + v ( β + s ( x ) ) , 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.