Spectral properties of ordinary differential operators admitting special decompositions

We investigate spectral properties of ordinary differential operators related to expressions of the form $ D^{\epsilon}+a $. Here $ a\in \mathbb{R} $ and $ D^{\epsilon} $ denotes a composition of $ \mathfrak{d} $ and $ \mathfrak{d}^+ $ according to the signs in the multi-index $ {\epsilon} $, where $ \mathfrak{d} $ is a first order linear differential expression, called delta-derivative, and $ \mathfrak{d}^+ $ is its formal adjoint in an appropriate $ L^2 $ space. In particular, Sturm-Liouville operators that admit the decomposition of the type $ \mathfrak{d}^+\mathfrak{d}+a $ are considered. We propose an approach, based on weak delta-derivatives and delta-Sobolev spaces, which is particularly useful in the study of the operators $ D^{\epsilon}+a $. Finally we examine a number of examples of operators, which are of the relevant form, naturally arising in analysis of classical orthogonal expansions.