Abdi Tadesse (PhD)Mesfin Yonas2021-02-242023-11-042021-02-242023-11-042021-01-21http://etd.aau.edu.et/handle/123456789/25216In this thesis, certain in_nite subsets of the set of positive integers are inves- tigated as possible spectra of regular weighted Sturm-Liouville eigenvalue problem with separated homogeneous boundary conditions. With the (con- ditional) exception of the set of square integers, it is shown that all the sets considered herein are not spectra of such a problem. We also state three necessary conditions in order for a set to be a spectrum of such eigenvalue problems. Concepts adopted from the area of study in Mathematical anal- ysis known as Asymptotic analysis will _gure prominently in the proofs of the main results.enSpectrumRegular WeightedSturm-Liouville ProblemsSet of IntegersThesis