Asymptotics of Solutions of the Sturm–Liouville Equation with Respect to a Parameter - Gomilko, A. M.; Pivovarchik, V. N. (Ukrains’kyi Matematychnyi Zhurnal, 2001)
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.