scholarly journals Spectral Properties of a Non-Self-Adjoint Differential Operator with Block-Triangular Operator Coefficients

2021 ◽  
Author(s):  
Aleksandr Kholkin
Keyword(s):  
Asymptotic Behavior ◽  
Differential Operator ◽  
Liouville Equation ◽  
Spectral Properties ◽  
Sufficient Conditions ◽  
Triangular Matrix ◽  
Spectral Singularities ◽  
Matrix Potential ◽  
Triangular Operator ◽  
Sturm Liouville

In this chapter, the Sturm-Liouville equation with block-triangular, increasing at infinity operator potential is considered. A fundamental system of solutions is constructed, one of which decreases at infinity, and the second increases. The asymptotic behavior at infinity was found out. The Green’s function and the resolvent for a non-self-adjoint differential operator are constructed. This allows to obtain sufficient conditions under which the spectrum of this non-self-adjoint differential operator is real and discrete. For a non-self-adjoint Sturm-Liouville operator with a triangular matrix potential growing at infinity, an example of operator having spectral singularities is constructed.

Sturm-Liouville operators with an indefinite weight function

1977 ◽  
Vol 78 (1-2) ◽  
pp. 161-191 ◽  
Author(s):  
K. Daho ◽  
H. Langer
Keyword(s):  
Weight Function ◽  
Liouville Equation ◽  
Spectral Properties ◽  
Scalar Product ◽  
Main Tool ◽  
Indefinite Weight ◽  
Indefinite Weight Function ◽  
Sturm Liouville ◽  
Liouville Operators

SynopsisSpectral properties of the singular Sturm-Liouville equation –(p−1y′)′ + qy = λry with an indefinite weight function r are studied in . The main tool is the theory of definitisable operators in spaces with an indefinite scalar product.

scholarly journals Inverse problem for Sturm-Liouville operators on a curve

2019 ◽  
Vol 50 (3) ◽  
pp. 349-359
Author(s):  
Andrey Aleksandrovich Golubkov ◽  
Yulia Vladimirovna Kuryshova
Keyword(s):  
Inverse Problem ◽  
Asymptotic Behavior ◽  
Transfer Matrix ◽  
Liouville Equation ◽  
Inverse Spectral Problem ◽  
Rectifiable Curve ◽  
Uniqueness Of The Solution ◽  
Sturm Liouville ◽  
Discontinuity Conditions ◽  
Liouville Operators

he inverse spectral problem for the Sturm-Liouville equation with a piecewise-entire potential function and the discontinuity conditions for solutions on a rectifiable curve \(\gamma \subset \textbf{C}\) by the transfer matrix along this curve is studied. By the method of a unit transfer matrix the uniqueness of the solution to this problem is proved with the help of studying of the asymptotic behavior of the solutions to the Sturm-Liouville equation for large values of the spectral parameter module.

scholarly journals Resolvent for Non-Self-Adjoint Differential Operator with Block-Triangular Operator Potential

2016 ◽  
Vol 2016 ◽  
pp. 1-6
Author(s):  
Aleksandr Mikhailovich Kholkin
Keyword(s):  
Differential Operator ◽  
Sufficient Conditions ◽  
Operator Potential ◽  
Triangular Operator

A resolvent for a non-self-adjoint differential operator with a block-triangular operator potential, increasing at infinity, is constructed. Sufficient conditions under which the spectrum is real and discrete are obtained.

scholarly journals Inverse problems for the Sturm–Liouville equation with a spectral parameter in the boundary condition

2017 ◽  
Author(s):  
Namig J. Guliyev
Keyword(s):  
Boundary Condition ◽  
Inverse Problems ◽  
Boundary Conditions ◽  
Spectral Parameter ◽  
Liouville Equation ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Eigenvalue Parameter ◽  
Sturm Liouville ◽  
Necessary And Sufficient

Inverse problems of recovering the coefficients of Sturm--Liouville problems with the eigenvalue parameter linearly contained in one of the boundary conditions are studied: (1) from the sequences of eigenvalues and norming constants; (2) from two spectra. Necessary and sufficient conditions for the solvability of these inverse problems are obtained.

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 The nonlocal problem with multi- point perturbations of the boundary conditions of the Sturm-type for an ordinary differential equation with involution of even order

2020 ◽  
Vol 54 (1) ◽  
pp. 64-78 ◽  
Author(s):  
Ya.O. Baranetskij ◽  
P.I. Kalenyuk ◽  
M. I. Kopach ◽  
A.V. Solomko
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Differential Operator ◽  
Boundary Conditions ◽  
Spectral Properties ◽  
Nonlocal Problem ◽  
Sufficient Conditions ◽  
Adjoint Problem ◽  
Multipoint Problem ◽  
Even Order

The spectral properties of the nonself-adjoint problem with multipoint perturbations of the Dirichlet conditions for differential operator of order $2n$ with involution are investigated. The system of eigenfunctions of a multipoint problem is constructed. Sufficient conditions have been established, under which this system is complete and, under some additional assumptions, forms the Riesz basis. The research is structured as follows. In section 2 we investigate the properties of the Sturm-type conditions and nonlocal problem with self-adjoint boundary conditions for the equation $$(-1)^ny^{(2n)}(x)+ a_{0}y^{(2n-1)}(x)+ a_{1}y^{(2n-1)}(1-x)=f(x),\,x\in (0,1).$$ In section 3 we study the spectral properties for nonlocal problem with nonself-adjoint boundary conditions for this equation. In sections 4 we construct a commutative group of transformation operators. Using spectral properties of multipoint problem and conditions for completeness the basis properties of the systems of eigenfunctions are established in section 5. In section 6 some analogous results are obtained for multipoint problems generated by differential equations with an involution and are proved the main theorems.

scholarly journals Spectral properties of the Klein-Gordons-wave equation with spectral parameter-dependent boundary condition

2004 ◽  
Vol 2004 (27) ◽  
pp. 1437-1445
Author(s):  
Gülen Başcanbaz-Tunca
Keyword(s):  
Boundary Condition ◽  
Wave Equation ◽  
Differential Operator ◽  
Finite Number ◽  
Spectral Parameter ◽  
Spectral Properties ◽  
Spectral Singularities ◽  
Klein Gordon ◽  
Parameter Dependent ◽  
Complex Valued

We investigate the spectrum of the differential operatorLλdefined by the Klein-Gordons-wave equationy″+(λ−q(x))2y=0,x∈ℝ+=[0,∞), subject to the spectral parameter-dependent boundary conditiony′(0)−(aλ+b)y(0)=0in the spaceL2(ℝ+), wherea≠±i,bare complex constants,qis a complex-valued function. Discussing the spectrum, we prove thatLλhas a finite number of eigenvalues and spectral singularities with finite multiplicities if the conditionslimx→∞q(x)=0,supx∈R+{exp(ϵx)|q′(x)|}<∞,ϵ>0, hold. Finally we show the properties of the principal functions corresponding to the spectral singularities.

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