On the General Ordinary Quasi-differential Operators with their\(L_{w}^{2}-\)Solutions and their Spectra

Abstract

In this chapter, we look at the general ordinary quasi-differential expression  of order  with complex coefficients and its formal adjoint \(\tau{^+}\) on the interval \((a,b)\). We will explain in the situation of one unique end-point and under proper conditions that all solutions of a general ordinary quasi-differential equation \((\tau\)- \(\lambda\)w)u = wf  are in the weighted Hilbert space \(L_{w}^{2}\)(a,b) provided that all solutions of the equations \((\tau\)- \(\lambda\)w)u = 0 and its adjoint \(\left(\tau^{+}-\bar{\lambda} w\right) v=0\) are in \(L_{w}^{2}\)(a,b) . It is also possible to derive a variety of results relating to the position of the point spectra and regularity fields of the operators created by such expressions. Some of these findings are expansions or generalizations of those found in the symmetric situation, while others are completely novel.