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.

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, ${1 \over 2} < \delta < 1$

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.

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 A note on the spectrum of discrete Klein-Gordon s-wave equation with eigenparameter dependent boundary condition

Filomat ◽  
2019 ◽  
Vol 33 (2) ◽  
pp. 449-455 ◽  
Author(s):  
Nimet Coskun ◽  
Nihal Yokus
Keyword(s):  
Boundary Condition ◽  
Wave Equation ◽  
Boundary Value Problem ◽  
Gordon Equation ◽  
S Wave ◽  
Klein Gordon ◽  
Complex Sequences ◽  
Quantitative Properties ◽  
Klein Gordon Equation ◽  
Jost Solution

This paper is concerned with the boundary value problem (BVP) for the discrete Klein-Gordon equation ?(an-1?yn-1)+(vn-?)2 yn = 0; n ? N and the boundary condition (?0+?1?)y1+(?0+?1)y0 = 0 where (an),(vn) are complex sequences, ?i, ?i ? C, i=0,1 and ? is a eigenparameter. The paper presents Jost solution, eigenvalues, spectral singularities and states some theorems concerning quantitative properties of the spectrum of this BVP under the condition ?n?N exp(?n?)(|1-an| + |vn|) < ? for ? > 0 and 1/2 ? ? ? 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 analysis of discrete dirac equation with generalized eigenparameter in boundary condition

Filomat ◽  
2019 ◽  
Vol 33 (18) ◽  
pp. 6039-6054 ◽  
Author(s):  
Turhan Koprubasi ◽  
Ram Mohapatra
Keyword(s):  
Boundary Condition ◽  
Spectral Analysis ◽  
Dirac Equation ◽  
Dirac Operator ◽  
Spectral Properties ◽  
Difference Operators ◽  
Spectral Singularities ◽  
First Order ◽  
Complex Sequences

Let L denote the discrete Dirac operator generated in ?2 (N,C2) by the non-selfadjoint difference operators of first order (an+1y(2)n+1 + bny(2)n + pny(1)n = ?y(1)n, an-1y(1)n-1 + bny(1)n + qny(2)n = ?y(2)n, n ? N, (0.1) with boundary condition Xp k=0 (y(2)1?k + y(1)0 ?k)?k=0, (0.2) where (an), (bn), (pn) and (qn), n ? N are complex sequences, ?i; ?i ? C, i = 0, 1, 2,..., p and ? is a eigenparameter. We discuss the spectral properties of L and we investigate the properties of the spectrum and the principal vectors corresponding to the spectral singularities of L, if ?? n=1 |n|(|1-an| + |1+bn| + |pn| + |qn|) < ? holds.

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