The Expansion Theorems for Sturm-Liouville Operators with an Involution Perturbation

2021 ◽  
Author(s):  
Abdizhahan Sarsenbi
Keyword(s):  
Differential Operator ◽  
Differential Operators ◽  
Second Order ◽  
Order Differential Operator ◽  
Spectral Expansions ◽  
Sturm Liouville ◽  
Complex Valued ◽  
Liouville Operators

In this work, we studied the Green’s functions of the second order differential operators with involution. Uniform equiconvergence of spectral expansions related to the second-order differential operators with involution is obtained. Basicity of eigenfunctions of the second-order differential operator operator with complex-valued coefficient is established.

An inverse problem for Sturm–Liouville operators on the half-line with complex weights

2019 ◽  
Vol 27 (3) ◽  
pp. 439-443
Author(s):  
Vjacheslav Yurko
Keyword(s):  
Inverse Problem ◽  
Inverse Problems ◽  
Differential Operators ◽  
Spectral Characteristics ◽  
Half Line ◽  
Sturm Liouville ◽  
The Given ◽  
Complex Valued ◽  
The Uniqueness Theorem ◽  
Liouville Operators

Abstract Second order differential operators on the half-line with complex-valued weights are considered. Properties of spectral characteristics are established, and the inverse problem of recovering operator’s coefficients from the given Weyl-type function is studied. The uniqueness theorem is proved for this class of nonlinear inverse problems, and a number of examples are provided.

Oscillation criteria of Nehari-type for Sturm–Liouville operators and elliptic differential operators of second order and the lower spectrum

1988 ◽  
Vol 109 (1-2) ◽  
pp. 127-144 ◽  
Author(s):  
F. Fiedler
Keyword(s):  
Differential Equation ◽  
Differential Operators ◽  
Second Order ◽  
Oscillation Criteria ◽  
Elliptic Differential Equations ◽  
Ordinary Differential Operators ◽  
Elliptic Differential Operators ◽  
Sturm Liouville ◽  
Lower Spectrum ◽  
Liouville Operators

SynopsisSufficient oscillation criteria of Nehari-type are established for the differential equation −uʺ(t) + q(t)u(t) = 0, 0<t<∞, with and without sign restrictions on q(t), respectively. These results are extended to Sturm-Liouville equations and elliptic differential equations of second order.In Section 7 we present conclusions for the lower spectrum of elliptic differential operators and also for the discreteness of the spectrum of certain ordinary differential operators of second order.

Reducibility of differential operators some: examples

1979 ◽  
Vol 86 (1) ◽  
pp. 29-33 ◽  
Author(s):  
B. Fishel ◽  
N. Denkel
Keyword(s):  
Differential Operator ◽  
Differential Operators ◽  
Volterra Operator ◽  
Second Order ◽  
Order Differential Operator ◽  
Order Operator ◽  
Minimal Operator ◽  
Deficiency Indices ◽  
First Order ◽  
Symmetric Operators

A symmetric operator on a Hilbert space, with deficiency indices (m; m) has self-adjoint extensions. These are ‘highly reducible’. The original operator may be irreducible, (see example (i), below). Can the mechanism whereby reducibility is achieved be understood? The concrete examples most readily studied are those associated with differential operators. It is easy to obtain operators, associated with a formal linear differential operator, having deficiency indices (m; m). What of reducibility? Nothing seems to be known. In the case of the first-order operator we were able, using the Volterra operator, to establish irreducibility of the associated minimal operator. To investigate symmetric operators associated with a second-order differential operator, different methods had to be developed. They apply also to the first-order operator, and we employ them to demonstrate the irreducibility of the associated minimal operator. In the second-order case the minimal operator proves reducible, and we also exhibit examples of reducibility of associated symmetric operators. It would clearly be of interest to elucidate the influence of the boundary conditions on reducibility.

scholarly journals On Basis Property of Root Functions For a Class Second Order Differential Operator

2020 ◽  
Vol 5 (1) ◽  
pp. 361-368
Author(s):  
Volkan Ala ◽  
Khanlar R. Mamedov
Keyword(s):  
Differential Operator ◽  
Boundary Conditions ◽  
Spectral Parameter ◽  
Liouville Equation ◽  
Basis Property ◽  
Second Order ◽  
Order Differential Operator ◽  
Root Functions ◽  
Sturm Liouville

AbstractIn this work we investigate the completeness, minimality and basis properties of the eigenfunctions of one class discontinuous Sturm-Liouville equation with a spectral parameter in boundary conditions.

scholarly journals Equiconvergence property for spectral expansions related to perturbations of the operator - u''(-x) with initial data

Filomat ◽  
2018 ◽  
Vol 32 (3) ◽  
pp. 1069-1078 ◽  
Author(s):  
Leonid Kritskov ◽  
Abdizhahan Sarsenbi
Keyword(s):  
Spectral Analysis ◽  
Initial Data ◽  
Differential Operators ◽  
Second Order ◽  
Spectral Expansions ◽  
Root Functions ◽  
Operator Form ◽  
Complex Valued

Uniform equiconvergence of spectral expansions related to the second-order differential operators with involution: -u''(-x) and -u''(-x) + q(x)u(x) with the initial data u(-1) = 0, u'(-1) = 0 is obtained. Starting with the spectral analysis of the unperturbed operator, the estimates of the Green?s functions are established and then applied via the contour integrating approach to the spectral expansions. As a corollary, it is proved that the root functions of the perturbed operator form the basis in L2(-1,1) for any complex-valued coefficient q(x) ? L2(-1,1).

Stability of Solutions of Differential-operator and Operator-difference Equations in the Sense of Perturbation of Operators

2006 ◽  
Vol 6 (3) ◽  
pp. 269-290 ◽  
Author(s):  
B. S. Jovanović ◽  
S. V. Lemeshevsky ◽  
P. P. Matus ◽  
P. N. Vabishchevich
Keyword(s):  
Differential Operator ◽  
Difference Equations ◽  
Hyperbolic Equations ◽  
Second Order ◽  
Difference Schemes ◽  
Order Differential Operator ◽  
Stability Of Solutions ◽  
Operator Equations ◽  
The Difference ◽  
Coefficient Stability

Abstract Estimates of stability in the sense perturbation of the operator for solving first- and second-order differential-operator equations have been obtained. For two- and three-level operator-difference schemes with weights similar estimates hold. Using the results obtained, we construct estimates of the coefficient stability for onedimensional parabolic and hyperbolic equations as well as for the difference schemes approximating the corresponding differential problems.

Sign in / Sign up

Export Citation Format

Share Document