Multiple Completeness of the Root Functions for a Certain Class of Pencils of Ordinary Differential Operators

Abstract

A class of polynomial pencils of ordinary differential operators with constant coefficients is considered in the article. The pencils from this class are generated by the n-th order ordinary differential expression on a finite segment (main segment) and two-point boundary conditions (at the ends of the segments). Coefficients of the differential expression are supposed to be polynomials of the spectral parameter of a special form with constant coefficients. The boundary conditions are supposed to depend on the spectral parameter polynomially. l (\({1\leq l \leq n-1}\)) of the boundary conditions are considered only at one end of the main segment. It is assumed that the roots of the characteristic equation of the pencils from this class are simple, non-zero and lie on two rays (or one as a special case) emanating from the origin. The author investigates m-fold (\({1\leq m \leq n}\)) completeness of the root functions of the pencils from this class in the space of square summable functions on the main segment. Sufficient conditions of the m-fold completeness of the root functions are obtained. The main idea of the proof of the theorem is a new asymptotics of the characteristic determinant of the pencil. The presented results supplement previous results of the author.