Aysmptotic estimates for eigenvalue problems having singularities

In this study, asymptotics approximations for the eigenvalues 𝜆𝑛 are obtained in the case of Dirichlet and Neumann boundary conditions satisfied by the second order differential equations y′′ + (𝜆 − 𝑞)𝑦 = 0, − ∞ < 𝑎 < 𝑥 < 𝑏 < ∞ in which 𝑞(𝑥) has singularities at two points in the interval (𝑎, 𝑏). In the first part, the basic results related to the Sturm Liouville theory and some examples are given. In the second part using Prüfer transformation and some other transformations together with the similar results presented in [1],[2] are used to obtain asymptotics approximations. Finally, the obtained results and suggestions for further improvements are provided.