scholarly journals On a Problem that Does not Have Basis Property of Root Vectors, Associated with a Perturbed Regular Operator of Multiple Differentiation

2020 ◽  
pp. 568-573
Author(s):  
Nurlan S. Imanbaev
Keyword(s):  
Spectral Problem ◽  
Boundary Value ◽  
Basis Property ◽  
Differentiation Operator ◽  
Boundary Value Condition ◽  
Root Functions ◽  
Multiple Differentiation ◽  
Strongly Regular ◽  
Integral Perturbation ◽  
System Of Root Functions

A spectral problem for a multiple differentiation operator with integral perturbation of boundary value conditions which are regular but not strongly regular is considered in the paper. The feature of the problem is the absence of the basis property of the system of root vectors. A characteristic determinant of the spectral problem is constructed. It is shown that absence of the basis property of the system of root functions of the problem is unstable with respect to the integral perturbation of the boundary value condition

NON STABILITY ON BASIS PROPERTY OF SYSTEMS ROOT VECTORS OF A LOADED MULTIPLE DIFFERENTIATION OPERATOR

2020 ◽  
pp. 78-83
Author(s):  
N.S. Imanbaev ◽  
Keyword(s):  
Differential Operators ◽  
Boundary Value ◽  
Basis Property ◽  
Differentiation Operator ◽  
A Value ◽  
Multiple Differentiation ◽  
Associated Functions ◽  
Strongly Regular ◽  
Small Change ◽  
Loaded Term

In this paper we consider perturbations of a second order differential equation of the spectral problem with a loaded term, containing a value of the unknown function at the point zero, with regular, but not strongly regular boundary value conditions. Question about basis property of eigen functions and associated functions (E&AF) systems of a loaded multiple differentiation operator is studied. In the case of non-self-adjoint ordinary differential operators, the basis property of systems of eigen functions and associated functions (E&AF), in addition to the boundary value conditions, can be affected by values of coefficients of the differential operator. Moreover, it is known that the basic properties of E&AF can be changed at a small change of values of the coefficients. This fact was first noted in Il’in V.A. In this paper problem non stability on basis property of systems root vectors of a loaded multiple differentiation operator.

scholarly journals Uniform Basis Property of the System of Root Vectors of the Dirac Operator

2018 ◽  
Vol 64 (1) ◽  
pp. 180-193
Author(s):  
A M Savchuk ◽  
I V Sadovnichaya
Keyword(s):  
Dirac Operator ◽  
Finite Number ◽  
Basis Property ◽  
One Dimensional ◽  
Uniform Estimates ◽  
Root Functions ◽  
Strongly Regular ◽  
System Of Root Functions ◽  
Complex Valued ◽  
Summable Potential

We study one-dimensional Dirac operator L on the segment [0,π] with regular in the sense of Birkhoff boundary conditions U and complex-valued summable potential P=(pij(x)), i,j=1,2. We prove uniform estimates for the Riesz constants of systems of root functions of a strongly regular operator L assuming that boundary-value conditions U and the number ∫(p1(x)-p4(x))dx are fixed and the potential P takes values from the ball B(0,R) of radius R in the space Lϰ for ϰ>1. Moreover, we can choose the system of root functions so that it consists of eigenfunctions of the operator L except for a finite number of root vectors that can be uniformly estimated over the ball ∥P∥ϰ≤R.

scholarly journals On Stability of Basis Property of Root Vectors System of the Sturm-Liouville Operator with an Integral Perturbation of Conditions in Nonstrongly Regular Samarskii-Ionkin Type Problems

2015 ◽  
Vol 2015 ◽  
pp. 1-6 ◽  
Author(s):  
N. S. Imanbaev
Keyword(s):  
Boundary Conditions ◽  
Liouville Operator ◽  
Basis Property ◽  
Differentiation Operator ◽  
Stability And Instability ◽  
Associated Functions ◽  
Sturm Liouville ◽  
Type Boundary ◽  
Eigenfunctions And Associated Functions ◽  
Integral Perturbation

We study a question on stability and instability of basis property of system of eigenfunctions and associated functions of the double differentiation operator with an integral perturbation of Samarskii-Ionkin type boundary conditions.

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