Asymptotics of the solutions of the Sturm–Liouville equation with singular coefficients

Abstract

We obtain asymptotic representations as λ→∞in the upper and lower half-planes for the solutions of the Sturm–Liouville equation

$$- y'' + p\left( x \right)y' + q\left( x \right)y = {\lambda ^2}p\left( x \right)y$$

, x ∈ [a,b] ⊂ R under the condition that q is a distribution of first-order singularity, ρ is a positive absolutely continuous function, and p belongs to the space L ₂[a, b].