Asymptotics of Solutions of the Sturm–Liouville Equation with Respect to a Parameter

Abstract

On a finite segment [0, l], we consider the differential equation

$$\left( {a\left( x \right)y\prime \left( x \right)} \right)\prime + \left[ {{\mu \rho }_{\text{1}} \left( x \right) + {\rho }_{2} \left( x \right)} \right]y\left( x \right) = 0$$

with a parameter μ ∈ C. In the case where a(x), ρ(x) ∈ L _∞[0, l], ρ _j (x) ∈ L ₁[0, l], j = 1, 2, a(x) ≥ m ₀ > 0 and ρ(x) ≥ m ₁ > 0 almost everywhere, and a(x)ρ(x) is a function absolutely continuous on the segment [0, l], we obtain exponential-type asymptotic formulas as $\left| {\mu } \right| \to \infty$ for a fundamental system of solutions of this equation.