A new approach to asymptotic formulas for eigenfunctions of discontinuous non-selfadjoint Sturm–Liouville operators

Abstract

In the present paper, boundary value problems for discontinuous non-selfadjoint Sturm–Liouville operators on a finite interval with boundary conditions nonlinearly dependent on the spectral parameter are considered, and a new approach for studying the asymptotic representation of the eigenfunctions and their partial derivatives is presented. We obtain the asymptotic representation of the solutions and the eigenvalue, and study some of their main properties. Then, we provide a constructive procedure to obtain the asymptotic form of the eigenfunctions and their partial derivatives in discontinuous case by the canonical form of the Bessel functions $$J_{\frac{1}{2}}(z)$$ , $$J_{\frac{3}{2}}(z)$$ and their derivatives.