Master's Thesis from the year 2016 in the subject Mathematics - Algebra, , language: English, abstract: In this thesis, using asymptotic integration, we have investigated the asymptotic of the eigensolutions and the deficiency indices of fourth order differential operators with unbounded coefficients as well as the location of absolutely continuous spectrum of self-adjoint extension operators. We have mainly endeavored to compute eigenvalues of fourth order differential operators when the coefficients are unbounded, determine the deficiency indices of such differential operator and the location of the absolutely continuous spectrum of the self-adjoint extension operator together with their spectral multiplicity. Results obtained for deficiency indices were in the range (2, 2) ≤ defT ≤ (4, 4) under different growth and decay conditions of co-efficients. The concept of unbounded operators provides an abstract framework for dealing with differential operators and unbounded observable such as in quantum mechanics. The theory of unbounded operators was developed by John Von Neumann in the late 1920s and early 1930s in an effort to solve problems related to quantum mechanics and other physical observables. This has provided the background on which other scholars have developed their work in differential operators. Higher order differential operators as defined on Hilbert spaces have received much attention though there still lays the problem of computing the eigenvalues of these higher order operators when the coefficients are unbounded.