Derivation of Asymptotic Formulas for Eigenvalues and Eigenfunctions Using the Ray Method

1972 ◽  
pp. 66-96
Author(s):  
Vasili M. Babič ◽  
Vladimir S. Buldyrev
Keyword(s):  
Asymptotic Formulas ◽  
Ray Method ◽  
Eigenvalues And Eigenfunctions

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.

Refined asymptotic formulas for eigenvalues and eigenfunctions of the Dirac system

2012 ◽  
Vol 85 (2) ◽  
pp. 240-242 ◽  
Author(s):  
M. Sh. Burlutskaya ◽  
V. P. Kurdyumov ◽  
A. P. Khromov
Keyword(s):  
Asymptotic Formulas ◽  
Dirac System ◽  
Eigenvalues And Eigenfunctions

scholarly journals Refinement Asymptotic Formulas of Eigenvalues and Eigenfunctions of a Fourth Order Linear Differential Operator with Transmission Condition and Discontinuous Weight Function

Symmetry ◽  
2019 ◽  
Vol 11 (8) ◽  
pp. 1060 ◽  
Author(s):  
Rando Rasul Qadir ◽  
Karwan Hama Faraj Jwamer
Keyword(s):  
Weight Function ◽  
Fourth Order ◽  
Asymptotic Expression ◽  
Linear Differential Operator ◽  
Fundamental Solutions ◽  
Transmission Condition ◽  
Asymptotic Formulas ◽  
Eigenvalues And Eigenfunctions ◽  
Order Linear ◽  
Linear Differential

In this paper, we promote the refinement method for estimating asymptotic expression of the fundamental solutions of a fourth order linear differential equation with discontinuous weight function and transmission conditions. These refinement solutions utilize more accurate asymptotic formulas for the eigenvalues and eigenfunctions for the problem.

scholarly journals On the Differential Operators with Periodic Matrix Coefficients

2009 ◽  
Vol 2009 ◽  
pp. 1-21 ◽  
Author(s):  
O. A. Veliev
Keyword(s):  
Differential Operator ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Ordinary Differential Equations ◽  
Differential Operators ◽  
The Self ◽  
Asymptotic Formulas ◽  
Eigenvalues And Eigenfunctions ◽  
Periodic Matrix ◽  
Matrix Coefficients

We obtain asymptotic formulas for eigenvalues and eigenfunctions of the operator generated by a system of ordinary differential equations with summable coefficients and quasiperiodic boundary conditions. Then by using these asymptotic formulas, we find conditions on the coefficients for which the number of gaps in the spectrum of the self-adjoint differential operator with the periodic matrix coefficients is finite.

scholarly journals ASYMPTOTIC DISTRIBUTION OF EIGENVALUES AND EIGENFUNCTIONS OF A NONLOCAL BOUNDARY VALUE PROBLEM

2021 ◽  
Vol 26 (2) ◽  
pp. 253-266
Author(s):  
Erdoğan Şen ◽  
Artūras Štikonas
Keyword(s):  
Boundary Condition ◽  
Boundary Value Problem ◽  
Asymptotic Distribution ◽  
Nonlocal Boundary Condition ◽  
Boundary Value ◽  
Nonlocal Boundary ◽  
Asymptotic Formulas ◽  
Nonlocal Boundary Value Problem ◽  
Eigenvalues And Eigenfunctions ◽  
Distribution Of Eigenvalues

In this work, we obtain asymptotic formulas for eigenvalues and eigenfunctions of the second order boundary-value problem with a Bitsadze–Samarskii type nonlocal boundary condition.

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.

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 The Modified Parseval Equality of Sturm-Liouville Problems with Transmission Conditions

2013 ◽  
Vol 2013 ◽  
pp. 1-11
Author(s):  
Mudan Bai ◽  
Jiong Sun ◽  
Siqin Yao
Keyword(s):  
Fundamental Solutions ◽  
Transmission Conditions ◽  
Asymptotic Formulas ◽  
Necessary Condition ◽  
Parseval Equality ◽  
Eigenvalues And Eigenfunctions ◽  
Sturm Liouville ◽  
Countably Infinite ◽  
Bounded Below ◽  
Sufficient And Necessary Condition

We consider the Sturm-Liouville (S-L) problems with very general transmission conditions on a finite interval. Firstly, we obtain the sufficient and necessary condition forλbeing an eigenvalue of the S-L problems by constructing the fundamental solutions of the problems and prove that the eigenvalues of the S-L problems are bounded below and are countably infinite. Furthermore, the asymptotic formulas of the eigenvalues and eigenfunctions of the S-L problems are obtained. Finally, we derive the eigenfunction expansion for Green's function of the S-L problems with transmission conditions and establish the modified Parseval equality in the associated Hilbert space.

Sign in / Sign up

Export Citation Format

Share Document