On the root functions of ordinary differential operators

The main objective of this thesis is to investigate the system of the root functions and the Riesz basis property of the second order non-self-adjoint Sturm-Liouville operator generated in L₂[0,1] by the differential expression l(y) = -y′′ + q(x) y where q is a complex-valued summable function on [0,1], and general regular boundary conditions that are not strongly regular. To this end, first we construct subtle asymptotic formulas for the eigenvalues and eigenfunctions of these operators for both cases q∈L₁[0,1] and q is an absolutely continuous function. Then using these formulas we find explicit conditions on the potential q such that the system of the root functions of the Sturm-Liouville operator with general regular boundary conditions does not form a Riesz basis. Also, we estimate the small eigenvalues of the second order non-self-adjoint Sturm-Liouville operators with general regular boundary conditions by the numerical methods. Finally, we give the error estimations and present some numerical examples. Key Words: Asymptotic formulas, Regular boundary conditions, Riesz basis, Numerical estimations of the eigenvalues.