scholarly journals Basis Properties of Eigenfunctions of Second-Order Differential Operators with Involution

2012 ◽  
Vol 2012 ◽  
pp. 1-6 ◽  
Author(s):  
Asylzat Kopzhassarova ◽  
Abdizhakhan Sarsenbi
Keyword(s):  
Differential Operator ◽  
Differential Operators ◽  
Second Order ◽  
Ordinary Differential Operator ◽  
Spectral Problems ◽  
Associated Functions ◽  
Eigenfunctions And Associated Functions

We study the basis properties of systems of eigenfunctions and associated functions for one kind of generalized spectral problems for a second-order ordinary differential operator.

A class of symmetric ordinary differential operators whose deficiency numbers differ by an integer†

1978 ◽  
Vol 82 (1-2) ◽  
pp. 117-134 ◽  
Author(s):  
Richard C. Gilbert
Keyword(s):  
Differential Operator ◽  
Positive Integer ◽  
Asymptotic Expansions ◽  
Differential Operators ◽  
Ordinary Differential Operator ◽  
Ordinary Differential Operators ◽  
Half Axis ◽  
Complex Coefficients

SynopsisFormulas are determined for the deficiency numbers of a formally symmetric ordinary differential operator with complex coefficients which have asymptotic expansions of a prescribed type on a half-axis. An implication of these formulas is that for any given positive integer there exists a formally symmetric ordinary differential operator whose deficiency numbers differ by that positive integer.

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.

scholarly journals Riesz Inequality for the System of Root Functions of Second Order Ordinary Differential Operator

2020 ◽  
Vol 19 ◽  
Keyword(s):  
Differential Operator ◽  
Sufficient Conditions ◽  
Second Order ◽  
Ordinary Differential Operator ◽  
Root Functions ◽  
Riesz Property ◽  
System Of Root Functions ◽  
The Given ◽  
Riesz Inequality

An ordinary differential operator of second order with coefficients is considered. The Riesz property of the system of root functions of the given operator is studied. The criterion of Bessel property in 2 L , of root functions system is established and use it to obtain sufficient conditions for the Riesz property of a system of normalized root functions of this operator in p L .

Bifurcation of Eigenvalues of Nonselfadjoint Differential Operators in Nonconservative Stability Problems

2002 ◽  
Author(s):  
Oleg N. Kirillov ◽  
Alexander P. Seyranian
Keyword(s):  
Parameter Space ◽  
Differential Operators ◽  
Eigenvalue Problems ◽  
Initial Point ◽  
Potential Force ◽  
Arbitrary Length ◽  
Eigen Values ◽  
Associated Functions ◽  
Linear Differential ◽  
Eigenfunctions And Associated Functions

In the present paper eigenvalue problems for non-selfadjoint linear differential operators smoothly dependent on a vector of real parameters are considered. Bifurcation of eigenvalues along smooth curves in the parameter space is studied. The case of multipleeigen value with Keldysh chain of arbitrary length is considered. Explicit expressions describing bifurcation of eigen-values are found. The obtained formulae use eigenfunctions and associated functions of the adjoint eigenvalue problems as well as the derivatives of the differential operator taken at the initial point of the parameter space. These results are important for the stability theory, sensitivity analysis and structural optimization. As a mechanical application the extended Beck’s problem of stability of an elastic column under action of potential force and tangential follower force is considered and discussed in detail.

Singular differential operators with spectra discrete and bounded below

1979 ◽  
Vol 84 (1-2) ◽  
pp. 117-134 ◽  
Author(s):  
Don B. Hinton ◽  
Roger T. Lewis
Keyword(s):  
Differential Operator ◽  
Differential Operators ◽  
Ordinary Differential Operator ◽  
Adjoint Operators ◽  
Bounded Below

SynopsisA weighted, formally self-adjoint ordinary differential operator l of order 2n is considered, and conditions are given on the coefficients of l which ensure that all self-adjoint operators associated with l have a spectrum which is discrete and bounded below. Both finite and infinite singularities are considered. The results are obtained by the establishment of certain conditions which imply that l is non-oscillatory.

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