scholarly journals Non-Self-Adjoint Singular Sturm-Liouville Problems with Boundary Conditions Dependent on the Eigenparameter

2010 ◽  
Vol 2010 ◽  
pp. 1-10
Author(s):  
Elgiz Bairamov ◽  
M. Seyyit Seyyidoglu
Keyword(s):  
Boundary Conditions ◽  
Finite Number ◽  
Analytic Functions ◽  
Liouville Problem ◽  
Uniqueness Theorems ◽  
Spectral Singularities ◽  
Sturm Liouville ◽  
Complex Valued

Let denote the operator generated in by the Sturm-Liouville problem: , , , where is a complex valued function and , with In this paper, using the uniqueness theorems of analytic functions, we investigate the eigenvalues and the spectral singularities of . In particular, we obtain the conditions on under which the operator has a finite number of the eigenvalues and the spectral singularities.

scholarly journals Spectral Properties of Nonhomogenous Differential Equations with Spectral Parameter in the Boundary Condition

2014 ◽  
Vol 22 (2) ◽  
pp. 109-120
Author(s):  
Özkan Karaman
Keyword(s):  
Boundary Value Problem ◽  
Spectral Parameter ◽  
Analytic Functions ◽  
Spectral Properties ◽  
Boundary Value ◽  
Uniqueness Theorems ◽  
Spectral Singularities ◽  
Boundary Properties ◽  
Boundary Uniqueness ◽  
Complex Valued

AbstractIn this paper, using the boundary properties of the analytic functions we investigate the structure of the discrete spectrum of the boundary value problem (0.1)$$\matrix{\hfill {iy_1^\prime + q_1 \left(x \right)y_2 - \lambda y_1 = \varphi _1 \left(x \right)\;\;} & \hfill {} \cr \hfill {- iy_2^\prime + q_2 \left(x \right)y_1 - \lambda y_2 = \varphi _2 \left(x \right),} & \hfill {x \in R_ + } \cr }$$ and the condition (0.2)$$\left({a_1 \lambda + b_1 } \right)y_2 \left({0,\lambda } \right) - \left({a_2 \lambda + b_2 } \right)y_1 \left({0,\lambda } \right) = 0$$ where q1,q2, φ1, φ2 are complex valued functions, ak ≠ 0, bk ≠ 0, k = 1, 2 are complex constants and λ is a spectral parameter. In this article, we investigate the spectral singularities and eigenvalues of (0.1), (0.2) using the boundary uniqueness theorems of analytic functions. In particular, we prove that the boundary value problem (0.1), (0.2) has a finite number of spectral singularities and eigenvalues with finite multiplicities under the conditions, $$\matrix{{\mathop {\sup }\limits_{x \in R_ + } \left[ {\left| {\varphi _k \left(x \right)} \right|\exp \left({\varepsilon x^\delta } \right)} \right] < \infty ,\;\;\;k = 1.2} \hfill \cr {\mathop {\sup }\limits_{x \in R_ + } \left[ {\left| {q_k \left(x \right)} \right|\exp \left({\varepsilon x^\delta } \right)} \right] < \infty ,\;\;\;k = 1.2} \hfill \cr }$$ for some ε > 0, ${1 \over 2} < \delta < 1$

scholarly journals Principal Functions of Non-Selfadjoint Sturm-Liouville Problems with Eigenvalue-Dependent Boundary Conditions

2011 ◽  
Vol 2011 ◽  
pp. 1-12 ◽  
Author(s):  
Nihal Yokuş
Keyword(s):  
Boundary Condition ◽  
Boundary Conditions ◽  
Differential Expression ◽  
Spectral Singularities ◽  
Sturm Liouville ◽  
Eigenvalue Dependent Boundary Conditions ◽  
Complex Valued

We consider the operator generated in by the differential expression , and the boundary condition , where is a complex-valued function and , with . In this paper we obtain the properties of the principal functions corresponding to the spectral singularities of .

scholarly journals Spectral Singularities of Sturm-Liouville Problems with Eigenvalue-Dependent Boundary Conditions

2009 ◽  
Vol 2009 ◽  
pp. 1-8 ◽  
Author(s):  
Elgiz Bairamov ◽  
Nihal Yokus
Keyword(s):  
Boundary Conditions ◽  
Liouville Equation ◽  
Spectral Singularities ◽  
Sturm Liouville ◽  
Eigenvalue Dependent Boundary Conditions ◽  
Complex Valued

LetLdenote the operator generated inL2(R+)by Sturm-Liouville equation−y′′+q(x)y=λ2y,x∈R+=[0,∞),y′(0)/y(0)=α0+α1λ+α2λ2, whereqis a complex-valued function andαi∈ℂ,i=0,1,2withα2≠0. In this article, we investigate the eigenvalues and the spectral singularities ofLand obtain analogs of Naimark and Pavlov conditions forL.

Spectral properties of Sturm–Liouville system with eigenvalue-dependent boundary conditions

2015 ◽  
Vol 26 (10) ◽  
pp. 1550080 ◽  
Author(s):  
Esra Kir Arpat ◽  
Gökhan Mutlu
Keyword(s):  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Finite Number ◽  
Spectral Parameter ◽  
Spectral Properties ◽  
Hermitian Matrix ◽  
Boundary Value ◽  
Spectral Singularities ◽  
Sturm Liouville ◽  
Eigenvalue Dependent Boundary Conditions

In this paper, we consider the boundary value problem [Formula: see text][Formula: see text] where λ is the spectral parameter and [Formula: see text] is a Hermitian matrix such that [Formula: see text] and αi ∈ ℂ, i = 0, 1, 2, with α2 ≠ 0. In this paper, we investigate the eigenvalues and spectral singularities of L. In particular, we prove that L has a finite number of eigenvalues and spectral singularities with finite multiplicities, under the Naimark and Pavlov conditions.

Sturm-Liouville Problem for Stationary Differential Operator with Nonlocal Two-Point Boundary Conditions

2006 ◽  
Vol 11 (1) ◽  
pp. 47-78 ◽  
Author(s):  
S. Pečiulytė ◽  
A. Štikonas
Keyword(s):  
Boundary Condition ◽  
Differential Operator ◽  
Boundary Conditions ◽  
Nonlocal Boundary Condition ◽  
Nonlocal Boundary ◽  
Liouville Problem ◽  
Complex Case ◽  
Qualitative Behavior ◽  
Point Boundary ◽  
Sturm Liouville

The Sturm-Liouville problem with various types of two-point boundary conditions is considered in this paper. In the first part of the paper, we investigate the Sturm-Liouville problem in three cases of nonlocal two-point boundary conditions. We prove general properties of the eigenfunctions and eigenvalues for such a problem in the complex case. In the second part, we investigate the case of real eigenvalues. It is analyzed how the spectrum of these problems depends on the boundary condition parameters. Qualitative behavior of all eigenvalues subject to the nonlocal boundary condition parameters is described.

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