Asymptotics of the Solutions of the Sturm--Liouville Equation with Singular Coefficients

V. E. Vladikina ; A. A. Shkalikov

Matematičeskie zametki, Tome 98 (2015) no. 6, pp. 832-841

Résumé

We obtain asymptotic representations as $\lambda \to \infty$ in the upper and lower half-planes for the solutions of the Sturm–Liouville equation $$ -y''+p(x)y'+q(x)y= \lambda ^2 \rho(x)y, \qquad x\in [a,b] \subset \mathbb{R}, $$ under the condition that $q$ is a distribution of first-order singularity, $\rho$ is a positive absolutely continuous function, and $p$ belongs to the space $L_2[a,b]$.

Détail

Mots-clés : Sturm–Liouville equation, singular coefficient
Keywords: asymptotic solution, Volterra integral operator, fundamental system of solutions, space of bounded functions.