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 2 R and D-epsilon denotes a composition of partial derivative and partial derivative(+) according to the signs in the multi-index epsilon, where d is a first order linear differential expression, called delta-derivative, and partial derivative(+) is its formal adjoint in an appropriate L-2 space. In particular, Sturm-Liouville operators that admit the decomposition of the type partial derivative(+)partial derivative + 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.