scholarly journals On the Spectrum of Odd Order Self-Adjoint Ordinary Differential Operators on the Real Line with Quasi-periodic Coefficients

1997 ◽  
Vol 135 (1) ◽  
pp. 41-65 ◽  
Author(s):  
B. Grébert ◽  
J.C. Guillot ◽  
F. Klopp
Keyword(s):  
Real Line ◽  
Differential Operators ◽  
Periodic Coefficients ◽  
The Real ◽  
Odd Order ◽  
Ordinary Differential Operators

Spectral Theory for the Differential Equation Lu= λ Mu

1958 ◽  
Vol 10 ◽  
pp. 431-446 ◽  
Author(s):  
Fred Brauer
Keyword(s):  
Differential Equation ◽  
Hilbert Space ◽  
Spectral Theory ◽  
Real Line ◽  
Differential Operators ◽  
Identity Operator ◽  
The Real ◽  
Ordinary Differential Operators ◽  
Extensive Bibliography

Let L and M be linear ordinary differential operators defined on an interval I, not necessarily bounded, of the real line. We wish to consider the expansion of arbitrary functions in eigenfunctions of the differential equation Lu = λMu on I. The case where M is the identity operator and L has a self-adjoint realization as an operator in the Hilbert space L 2(I) has been treated in various ways by several authors; an extensive bibliography may be found in (4) or (8).

The Two-Sided Factorization of Ordinary Differential Operators

1980 ◽  
Vol 32 (5) ◽  
pp. 1045-1057 ◽  
Author(s):  
Patrick J. Browne ◽  
Rodney Nillsen
Keyword(s):  
Differential Operator ◽  
Linear Function ◽  
Bilinear Form ◽  
Differential Operators ◽  
Real Numbers ◽  
The Real ◽  
Differentiable Functions ◽  
Ordinary Differential Operators ◽  
Image Position

Throughout this paper we shall use I to denote a given interval, not necessarily bounded, of real numbers and Cn to denote the real valued n times continuously differentiable functions on I and C0 will be abbreviated to C. By a differential operator of order n we shall mean a linear function L:Cn → C of the form1.1where pn(x) ≠ 0 for x ∊ I and pi ∊ Cj 0 ≦ j ≦ n. The function pn is called the leading coefficient of L.It is well known (see, for example, [2, pp. 73-74]) thai a differential operator L of order n uniquely determines both a differential operator L* of order n (the adjoint of L) and a bilinear form [·,·]L (the Lagrange bracket) so that if D denotes differentiation, we have for u, v ∊ Cn,1.2

Linear quasi-differential operators in locally integrable spaces on the real line

2000 ◽  
Vol 130 (4) ◽  
pp. 671-698 ◽  
Author(s):  
R. R. Ashurov ◽  
W. N. Everitt
Keyword(s):  
Banach Spaces ◽  
Real Line ◽  
Differential Operators ◽  
Linear Topological Space ◽  
Topological Spaces ◽  
The Real ◽  
Conjugate Space ◽  
Linear Differential ◽  
Definition Of ◽  
Locally Convex

The theory of ordinary linear quasi-differential expressions and operators has been extensively developed in integrable-square Hilbert spaces. There is also an extensive theory of ordinary linear differential expressions and operators in integrable-p Banach spaces.However, the basic definition of linear quasi-differential expressions involves Lebesgue locally integrable spaces on intervals of the real line. Such spaces are not Banach spaces but can be considered as complete locally convex linear topological spaces where the topology is derived from a countable family of semi-norms. The first conjugate space can also be defined as a complete locally convex linear topological space, but now with the topology derived as a strict inductive limit.This paper develops the properties of linear quasi-differential operators in a locally integrable space and the first conjugate space. Conjugate and preconjugate operators are defined in, respectively, dense and total domains.

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