ASYMPTOTIC FORMULAS FOR EIGENVALUES AND EIGENFUNCTIONS OF A NONSELFADJOINT STURM-LIOUVILLE OPERATOR

2009 ◽  
Author(s):  
KHANLAR R. MAMEDOV ◽  
HAMZA MENKEN
Keyword(s):  
Liouville Operator ◽  
Asymptotic Formulas ◽  
Eigenvalues And Eigenfunctions ◽  
Sturm Liouville

scholarly journals Asymptotic Behaviour of Eigenvalues and Eigenfunctions of a Sturm-Liouville Problem with Retarded Argument

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Erdoğan Şen ◽  
Jong Jin Seo ◽  
Serkan Araci
Keyword(s):  
Boundary Value Problem ◽  
Liouville Operator ◽  
Liouville Problem ◽  
Asymptotic Formulas ◽  
Retarded Argument ◽  
Eigenvalues And Eigenfunctions ◽  
Transmission Coefficients ◽  
Discontinuous Boundary ◽  
Sturm Liouville ◽  
Special Case

In the present paper, a discontinuous boundary-value problem with retarded argument at the two points of discontinuities is investigated. We obtained asymptotic formulas for the eigenvalues and eigenfunctions. This is the first work containing two discontinuities points in the theory of differential equations with retarded argument. In that special case the transmission coefficients and retarded argument in the results obtained in this work coincide with corresponding results in the classical Sturm-Liouville operator.

Sturm-Liouville Problems with Discontinuities at Two Interior Points

2018 ◽  
Vol 85 (1-2) ◽  
pp. 70
Author(s):  
Hongmei Han
Keyword(s):  
Green Function ◽  
Boundary Conditions ◽  
Liouville Operator ◽  
Resolvent Operator ◽  
Asymptotic Formulas ◽  
Eigenvalues And Eigenfunctions ◽  
Sturm Liouville ◽  
Basic Solutions ◽  
The Resolvent Operator ◽  
Interior Points

<p>In this paper, we study the Sturm-Liouville operator with eigenparameter-dependent boundary conditions and transmission conditions at two interior points. We establish a new operator <em>A</em> associated with the problem, prove the operator <em>A</em> is self-adjoint in an appropriate space <em>H</em>, construct the basic solutions and investigate some properties of the eigenvalues and corresponding eigenfunctions, then obtain asymptotic formulas for the eigenvalues and eigenfunctions, its Green function and the resolvent operator are also involved.</p>

scholarly journals A Sturm-Liouville Problem with a Discontinuous Coefficient and Containing an Eigenparameter in the Boundary Condition

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Erdoğan Şen
Keyword(s):  
Boundary Condition ◽  
Green Function ◽  
Boundary Conditions ◽  
Liouville Operator ◽  
Liouville Problem ◽  
Fundamental Solutions ◽  
Asymptotic Formulas ◽  
Eigenvalues And Eigenfunctions ◽  
Sturm Liouville ◽  
Interior Points

We study a Sturm-Liouville operator with eigenparameter-dependent boundary conditions and transmission conditions at two interior points. We give an operator-theoretic formulation, construct fundamental solutions, investigate some properties of the eigenvalues and corresponding eigenfunctions of the discontinuous Sturm-Liouville problem and then obtain asymptotic formulas for the eigenvalues and eigenfunctions and find Green function of the discontinuous Sturm-Liouville problem.

scholarly journals Fundamental Spectral Theory of Fractional Singular Sturm-Liouville Operator

2013 ◽  
Vol 2013 ◽  
pp. 1-7 ◽  
Author(s):  
Erdal Bas
Keyword(s):  
Spectral Theory ◽  
Liouville Operator ◽  
Spectral Properties ◽  
Liouville Problem ◽  
Eigenvalues And Eigenfunctions ◽  
Sturm Liouville

We give the theory of spectral properties for eigenvalues and eigenfunctions of Bessel type of fractional singular Sturm-Liouville problem. We show that the eigenvalues and eigenfunctions of the problem are real and orthogonal, respectively. Furthermore, we prove new approximations about the topic.

Partial inverse problems for the Sturm–Liouville operator on a star-shaped graph with mixed boundary conditions

2018 ◽  
Vol 26 (1) ◽  
pp. 1-12 ◽  
Author(s):  
Natalia Pavlovna Bondarenko
Keyword(s):  
Inverse Problems ◽  
Boundary Conditions ◽  
Liouville Operator ◽  
Mixed Boundary ◽  
Constructive Method ◽  
Asymptotic Formulas ◽  
Riesz Basis Property ◽  
Partial Inverse ◽  
Sturm Liouville ◽  
Shaped Graph

AbstractThe Sturm–Liouville operator on a star-shaped graph with different types of boundary conditions (Robin and Dirichlet) in different vertices is studied. Asymptotic formulas for the eigenvalues are derived and partial inverse problems are solved: we show that the potential on one edge can be uniquely determined by different parts of the spectrum if the potentials on the other edges are known. We provide a constructive method for the solution of the inverse problems, based on the Riesz basis property of some systems of vector functions.

scholarly journals Asymptotic Behavior of Solutions of the Dirac System with an Integrable Potential

2021 ◽  
Vol 93 (5) ◽  
Author(s):  
Łukasz Rzepnicki
Keyword(s):  
Asymptotic Behavior ◽  
Spectral Parameter ◽  
Asymptotic Formulas ◽  
Asymptotic Behavior Of Solutions ◽  
Dirac System ◽  
Eigenvalues And Eigenfunctions ◽  
Integrable Potential ◽  
Sturm Liouville ◽  
Complex Valued ◽  
Liouville Operators

AbstractWe consider the Dirac system on the interval [0, 1] with a spectral parameter $$\mu \in {\mathbb {C}}$$ μ ∈ C and a complex-valued potential with entries from $$L_p[0,1]$$ L p [ 0 , 1 ] , where $$1\le p$$ 1 ≤ p . We study the asymptotic behavior of its solutions in a strip $$|\mathrm{Im}\,\mu |\le d$$ | Im μ | ≤ d for $$\mu \rightarrow \infty $$ μ → ∞ . These results allow us to obtain sharp asymptotic formulas for eigenvalues and eigenfunctions of Sturm–Liouville operators associated with the aforementioned Dirac system.

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