Properties of Spectral Expansions Corresponding to Non-Self-Adjoint Differential Operators

Abstract

This paper is a survey of results in the spectral theory of differential operators generated by ordinary differential expressions and also by partial differential expressions of elliptic type. Our main focus is on the non-self-adjoint case. In contrast to the theory of self-adjoint differential operators, in which a firm foundation of functional analysis methods was laid due to the efforts of many mathematicians, in many respects, a universal conception of approaches to studying the problems arisen was created, and, finally much experience was accumulated in scientific publications for more than a century, the spectral theory of non-self-adjoint operators contains at present fairly many open problems. This does not mean at all that little has been done in this field: a list of all publications devoted to this theme, if it has ever been composed, would look at least like that in the theory of self-adjoint problems. All this is explained by the fact that often to study a new class of non-self-adjoint problems, we need to elaborate new methods using a “fine adjustment” of the functional analysis technique. Of course, the present survey does not claim to be an exhaustive presentation of scientific results and methods of the theory of non-self-adjoint differential operators. Here, we pay considerable attention to the studies carried out at the Chair of General Mathematics of the Department of Computational Mathematics and Cybernetics of the M.V. Lomonosov Moscow State University over a period of more than 30 years. They mainly concern one aspect or another of convergence of spectral expansions related to non-self-adjoint differential operators. The methodology elaborated there turns out to be a fairly effective tool for solving many new problems in this field. The authors try to make the reader familiar with the main results obtained up to now and give an idea of the methods elaborated for their proof. This specific character of the survey explains a certain “narrow specialization” and subjectiveness of the list of literature cited.