Some boundary properties of analytic functions

1954 ◽  
Vol 61 (1) ◽  
pp. 186-199 ◽  
Author(s):  
F. Bagemihl ◽  
W. Seidel
Keyword(s):  
Analytic Functions ◽  
Boundary Properties

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

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