Spectral properties of ordinary differential operators admitting special decompositions
The paper deals with 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 \(\partial\) and \(\partial^+\) according to the signs in the multi-index \(\epsilon\), where \(\partial\) is a first order linear differential expression, called delta-derivative, and \(\partial^+\) is its formal adjoint in an appropriate \(L_2\) space. In particular, Sturm-Liouville operators that admit the decomposition of the type \(\partial^+\partial + a\) are considered. The authors 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, they examine a number of examples of operators, which are of the relevant form, naturally arising in analysis of classical orthogonal expansions.
scientific article; zbMATH DE number 7359976
scientific article