Results Math 72 (2017), 281–301
2016 Springer International Publishing
published online August 31, 2016
Results in Mathematics
Multiple Completeness of the Root Functions
for a Certain Class of Pencils of Ordinary
Victor S. Rykhlov
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 ≤ l ≤ 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 ≤
m ≤ 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.
Mathematics Subject Classiﬁcation. Primary 34L10; Secondary 34B07,
Keywords. Pencil of ordinary diﬀerential operators, root functions,
multiple completeness, polynomial pencil of operators, system of root
functions, eigenfunctions, associated functions.
The results were obtained within the framework of the state task of Russian Ministry of
Education and Science (Project 1.1520.2014K).