On Multiple Completeness of Root Functions for Ordinary Differential Polynomial Pencils with Constant Coefficients
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ON MULTIPLE COMPLETENESS OF ROOT FUNCTIONS FOR ORDINARY DIFFERENTIAL POLYNOMIAL PENCILS WITH CONSTANT COEFFICIENTS V. S. Rykhlov
UDC 517.927.25
Abstract. In the space of square integrable functions on a finite segment, we consider a class of polynomial pencils of nth-order ordinary differential operators with constant coefficients and two-point boundary conditions (at the edges of the segment). We suppose that all roots of the characteristic equations of pencils of the said class are simple and nonzero. We find sufficient conditions for the m-multiple completeness (1 ≤ m ≤ n) of the system of root functions of pencils from the specified class in the space of square integrable functions on the said segment.
CONTENTS 1. Introduction . . . . . . . . . . . . . . . . . . . 1.1. Problem formulation . . . . . . . . . . . 1.2. Brief historical review . . . . . . . . . . . 1.3. Main results . . . . . . . . . . . . . . . . 2. Auxiliary Results and Estimate Lemma . . . 3. Completeness Theorems: Proofs . . . . . . . 3.1. The proofs of Theorems 1.1, 1.2, and 1.3 3.2. The proof of Theorem 1.4 . . . . . . . . 4. Conclusions . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . .
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683 683 684 685 688 697 699 701 703 703
Introduction
1.1. Problem formulation. In the space L2 [0, 1], consider the pencil of ordinary differential operators L(λ) generated (on the segment [0, 1]) by the differential expression (y, λ) := pn (x, λ)y (n) + pn−1 (x, λ)y (n−1) + · · · + p0 (x, λ)y
(1.1)
and the linearly independent boundary conditions Ui (y, λ) :=
n−1
αij (λ)y (j) (0) + βij (λ)y (j) (1) = 0,
i = 1, n,
(1.2)
j=0
where λ from C is the spectral parameter, pj (x, λ) =
n−j s=0
pjs (x)λs , pjs (x) ∈ L1 [0, 1], and αij (λ) and
βij (λ) are arbitrary polynomials with respect to λ. We use the known definitions of eigenvalues, eigenfunctions and adjoint functions (in brief, root functions), and Keldysh derived chains (see [6, 9]). Let Λ := {λk } be the set of all eigenvalues of the pencil L(λ) and Y := {yk } be the set of all root function of L(λ) corresponding to the set Λ. It is assumed that Λ is a countable set. Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 63, No. 2, Proceedings of the Crimean Autumn Mathematical School-Symposium, 2017. c 2020 Springer Science+Business Media, LLC 1072–3374/20/2504–0683
683
Definition 1.1. We say that the system Y of root functions of the pencil L(λ) is m-multiply comple
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