<abstract xmlns="http://www.w3.org/1999/xhtml">In this paper, using the boundary properties of the analytic functions we investigate the structure of the discrete spectrum of the boundary value problem
<disp-formula id="DF1_w2aab2b8b1b1b7b1aab1c14b1b1Aa"><label>(0.1)</label><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/auom-2014-0036_i1.jpg"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="right"><mtr columnalign="right"><mtd columnalign="right"><mrow><msubsup><mi mathvariant="italic">iy</mi><mn>1</mn><mo>'</mo></msubsup><mo>+</mo><msub><mi>q</mi><mn>1</mn></msub><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><msub><mi>y</mi><mn>2</mn></msub><mo>−</mo><mi>λ</mi><msub><mi>y</mi><mn>1</mn></msub><mo>=</mo><msub><mi>ϕ</mi><mn>1</mn></msub><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mo> </mo><mo> </mo></mrow></mtd><mtd columnalign="right"><mrow></mrow></mtd></mtr><mtr columnalign="right"><mtd columnalign="right"><mrow><mo>−</mo><msubsup><mi mathvariant="italic">iy</mi><mn>2</mn><mo>'</mo></msubsup><mo>+</mo><msub><mi>q</mi><mn>2</mn></msub><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><msub><mi>y</mi><mn>1</mn></msub><mo>−</mo><mi>λ</mi><msub><mi>y</mi><mn>2</mn></msub><mo>=</mo><msub><mi>ϕ</mi><mn>2</mn></msub><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><mo>,</mo></mrow></mtd><mtd columnalign="right"><mrow><mi>x</mi><mo>∈</mo><msub><mi>R</mi><mo>+</mo></msub></mrow></mtd></mtr></mtable></math><tex-math>$$\matrix{\hfill {iy_1^\prime + q_1 \left(x \right)y_2 - \lambda y_1 = \varphi _1 \left(x \right)\;\;} &amp; \hfill {} \cr \hfill {- iy_2^\prime + q_2 \left(x \right)y_1 - \lambda y_2 = \varphi _2 \left(x \right),} &amp; \hfill {x \in R_ + } \cr }$$</tex-math></alternatives></disp-formula>
and the condition
<disp-formula id="DF2_w2aab2b8b1b1b7b1aab1c14b1b3Aa"><label>(0.2)</label><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/auom-2014-0036_i2.jpg"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mrow><mo stretchy="false">(</mo><mrow><msub><mi>a</mi><mn>1</mn></msub><mi>λ</mi><mo>+</mo><msub><mi>b</mi><mn>1</mn></msub></mrow><mo stretchy="false">)</mo></mrow><msub><mi>y</mi><mn>2</mn></msub><mrow><mo stretchy="false">(</mo><mrow><mn>0</mn><mo>,</mo><mi>λ</mi></mrow><mo stretchy="false">)</mo></mrow><mo>−</mo><mrow><mo stretchy="false">(</mo><mrow><msub><mi>a</mi><mn>2</mn></msub><mi>λ</mi><mo>+</mo><msub><mi>b</mi><mn>2</mn></msub></mrow><mo stretchy="false">)</mo></mrow><msub><mi>y</mi><mn>1</mn></msub><mrow><mo stretchy="false">(</mo><mrow><mn>0</mn><mo>,</mo><mi>λ</mi></mrow><mo stretchy="false">)</mo></mrow><mo>=</mo><mn>0</mn></math><tex-math>$$\left({a_1 \lambda + b_1 } \right)y_2 \left({0,\lambda } \right) - \left({a_2 \lambda + b_2 } \right)y_1 \left({0,\lambda } \right) = 0$$</tex-math></alternatives></disp-formula>
where <italic>q</italic>1,<italic>q</italic>2, <italic>φ</italic>1, <italic>φ</italic>2 are complex valued functions, <italic>ak</italic> ≠ 0, <italic>bk</italic> ≠ 0, <italic>k</italic> = 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,
<disp-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/auom-2014-0036_i3.jpg"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="block"><mtable columnalign="left"><mtr columnalign="left"><mtd columnalign="left"><mrow><munder><mrow><mi>sup</mi><mo>⁡</mo></mrow><mrow><mi>x</mi><mo>∈</mo><msub><mi>R</mi><mo>+</mo></msub></mrow></munder><mrow><mo stretchy="false">[</mo><mrow><mrow><mo stretchy="false">|</mo><mrow><msub><mi>ϕ</mi><mi>k</mi></msub><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">|</mo></mrow><mi>exp</mi><mo>⁡</mo><mrow><mo stretchy="false">(</mo><mrow><mi>ε</mi><msup><mi>x</mi><mi>δ</mi></msup></mrow><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">]</mo></mrow><mo>&lt;</mo><mi>∞</mi><mo>,</mo><mo> </mo><mo> </mo><mo> </mo><mi>k</mi><mo>=</mo><mn>1.2</mn></mrow></mtd></mtr><mtr columnalign="left"><mtd columnalign="left"><mrow><munder><mrow><mi>sup</mi><mo>⁡</mo></mrow><mrow><mi>x</mi><mo>∈</mo><msub><mi>R</mi><mo>+</mo></msub></mrow></munder><mrow><mo stretchy="false">[</mo><mrow><mrow><mo stretchy="false">|</mo><mrow><msub><mi>q</mi><mi>k</mi></msub><mrow><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">|</mo></mrow><mi>exp</mi><mo>⁡</mo><mrow><mo stretchy="false">(</mo><mrow><mi>ε</mi><msup><mi>x</mi><mi>δ</mi></msup></mrow><mo stretchy="false">)</mo></mrow></mrow><mo stretchy="false">]</mo></mrow><mo>&lt;</mo><mi>∞</mi><mo>,</mo><mo> </mo><mo> </mo><mo> </mo><mi>k</mi><mo>=</mo><mn>1.2</mn></mrow></mtd></mtr></mtable></math><tex-math>$$\matrix{{\mathop {\sup }\limits_{x \in R_ + } \left[ {\left| {\varphi _k \left(x \right)} \right|\exp \left({\varepsilon x^\delta } \right)} \right] &lt; \infty ,\;\;\;k = 1.2} \hfill \cr {\mathop {\sup }\limits_{x \in R_ + } \left[ {\left| {q_k \left(x \right)} \right|\exp \left({\varepsilon x^\delta } \right)} \right] &lt; \infty ,\;\;\;k = 1.2} \hfill \cr }$$</tex-math></alternatives></disp-formula>
for some ε > 0, 
<inline-formula><alternatives><inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/auom-2014-0036_i4.jpg"></inline-graphic><math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>&lt;</mo><mi>δ</mi><mo>&lt;</mo><mn>1</mn></math><tex-math>${1 \over 2} &lt; \delta &lt; 1$</tex-math></alternatives></inline-formula></abstract>

In 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)iy1'+q1(x)y2−λy1=ϕ1(x)  −iy2'+q2(x)y1−λy2=ϕ2(x),x∈R+$$\matrix{\hfill {iy_1^\prime + q_1 \left(x \right)y_2 - \lambda y_1 = \varphi _1 \left(x \right)\;\;} & \hfill {} \cr \hfill {- iy_2^\prime + q_2 \left(x \right)y_1 - \lambda y_2 = \varphi _2 \left(x \right),} & \hfill {x \in R_ + } \cr }$$
and the condition
(0.2)(a1λ+b1)y2(0,λ)−(a2λ+b2)y1(0,λ)=0$$\left({a_1 \lambda + b_1 } \right)y_2 \left({0,\lambda } \right) - \left({a_2 \lambda + b_2 } \right)y_1 \left({0,\lambda } \right) = 0$$
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,
sup⁡x∈R+[|ϕk(x)|exp⁡(εxδ)]<∞,   k=1.2sup⁡x∈R+[|qk(x)|exp⁡(εxδ)]<∞,   k=1.2$$\matrix{{\mathop {\sup }\limits_{x \in R_ + } \left[ {\left| {\varphi _k \left(x \right)} \right|\exp \left({\varepsilon x^\delta } \right)} \right] < \infty ,\;\;\;k = 1.2} \hfill \cr {\mathop {\sup }\limits_{x \in R_ + } \left[ {\left| {q_k \left(x \right)} \right|\exp \left({\varepsilon x^\delta } \right)} \right] < \infty ,\;\;\;k = 1.2} \hfill \cr }$$
for some ε > 0, 
12<δ<1${1 \over 2} < \delta < 1$

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

Department of Mathematics, Faculty of Science and Letters, Kahramanmaras Sutcu Imam University, Avsar Campus,46100, K.Maras-Turkey.

Analele ştiinţifice ale Universităţii "Ovidius" Constanţa. Seria Matematică

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

In 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)iy1'+q1(x)y2−λy1=ϕ1(x)  −iy2'+q2(x)y1−λy2=ϕ2(x),x∈R+$$\matrix{\hfill {iy_1^\prime + q_1 \left(x \right)y_2 - \lambda y_1 = \varphi _1 \left(x \right)\;\;} & \hfill {} \cr \hfill {- iy_2^\prime + q_2 \left(x \right)y_1 - \lambda y_2 = \varphi _2 \left(x \right),} & \hfill {x \in R_ + } \cr }$$
and the condition
(0.2)(a1λ+b1)y2(0,λ)−(a2λ+b2)y1(0,λ)=0$$\left({a_1 \lambda + b_1 } \right)y_2 \left({0,\lambda } \right) - \left({a_2 \lambda + b_2 } \right)y_1 \left({0,\lambda } \right) = 0$$
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,
sup⁡x∈R+[|ϕk(x)|exp⁡(εxδ)] 0, 
12<δ<1${1 \over 2} < \delta < 1$

In this paper, using the boundary properties of the analytic functions we investigate the structure of the discrete spectrum of the boundary value...

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

Online veröffentlicht: 20. Okt. 2015

Seitenbereich: 109 - 120

Eingereicht: 01. Feb. 2013

Akzeptiert: 01. Feb. 2014

DOI: https://doi.org/10.2478/auom-2014-0036

Schlüsselwörter
Schrödinger equation, spectrum, spectral operators, singularity

© 2014 Özkan Karaman, published by De Gruyter Open

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

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

Online veröffentlicht: 20. Okt. 2015

Seitenbereich: 109 - 120

Eingereicht: 01. Feb. 2013

Akzeptiert: 01. Feb. 2014

DOI: https://doi.org/10.2478/auom-2014-0036

SchlüsselwörterSchrödinger equation, spectrum, spectral operators, singularity

© 2014 Özkan Karaman, published by De Gruyter Open

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

Schlüsselwörter
Schrödinger equation, spectrum, spectral operators, singularity