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

摘要

Asymptotic formulas as x →∞ are obtained for a fundamental system of solutions to equations of the form $$lleft( y ight): = {left( { - 1} ight)^n}{left( {pleft( x ight){y^{left( n ight)}}} ight)^{left( n ight)}} + qleft( x ight)y = lambda y,x in [1,infty )$$ 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 $$pleft( x ight) = {left( {1 + rleft( x ight)} ight)^{ - 1}},r in {L^1}left( {1,infty } ight)$$ 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 $$sigma in {L^1}left( {1,infty } ight)ifk < n,$$ σ ∈ L 1 ( 1 , ∞ ) i f k < n , $$left| sigma ight|left( {1 + left| r ight|} ight)left( {1 + left| sigma ight|} ight) in {L^1}left( {1,infty } ight)ifk = n$$ | σ | ( 1 + | r | ) ( 1 + | σ | ) ∈ L 1 ( 1 , ∞ ) i f k = n . Similar results are obtained for functions representable as $$pleft( x ight) = {x^{2n + v}}{left( {1 + rleft( x ight)} ight)^{ - 1}},q = {sigma ^{left( k ight)}},sigma left( x ight) = {x^{k + v}}left( {eta + sleft( x ight)} ight)$$ 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.