The spectrum of discrete Dirac operator with a general boundary condition
Abstract In this paper, we aim to investigate the spectrum of the nonselfadjoint operator L generated in the Hilbert space l 2 ( N , C 2 ) $l_{2}(\mathbb{N},\mathbb{C}^{2})$ by the discrete Dirac system { y n + 1 ( 2 ) − y n ( 2 ) + p n y n ( 1 ) = λ y n ( 1 ) , − y n ( 1 ) + y n − 1 ( 1 ) + q n y n...
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
SpringerOpen
2020-08-01
|
| Series: | Advances in Difference Equations |
| Subjects: | |
| Online Access: | http://link.springer.com/article/10.1186/s13662-020-02851-2 |
| _version_ | 1818300966972162048 |
|---|---|
| author | Nimet Coskun Nihal Yokus |
| author_facet | Nimet Coskun Nihal Yokus |
| author_sort | Nimet Coskun |
| collection | DOAJ |
| description | Abstract In this paper, we aim to investigate the spectrum of the nonselfadjoint operator L generated in the Hilbert space l 2 ( N , C 2 ) $l_{2}(\mathbb{N},\mathbb{C}^{2})$ by the discrete Dirac system { y n + 1 ( 2 ) − y n ( 2 ) + p n y n ( 1 ) = λ y n ( 1 ) , − y n ( 1 ) + y n − 1 ( 1 ) + q n y n ( 2 ) = λ y n ( 2 ) , n ∈ N , $$ \textstyle\begin{cases} y_{n+1}^{ (2 )} - y_{n}^{ (2 )} + p_{n} y_{n}^{ (1 )} =\lambda y_{n}^{ (1 )},\\ - y_{n}^{ (1 )} + y_{n-1}^{ (1 )} + q_{n} y_{n}^{ (2 )} =\lambda y_{n}^{ (2 )}, \end{cases}\displaystyle \quad n\in \mathbb{N}, $$ and the general boundary condition ∑ n = 0 ∞ h n y n = 0 , $$ \sum_{n = 0}^{\infty } h_{n}y_{n} = 0, $$ where λ is a spectral parameter, Δ is the forward difference operator, ( h n $h_{n}$ ) is a complex vector sequence such that h n = ( h n ( 1 ) , h n ( 2 ) ) $h_{n} = ( h_{n}^{(1)}, h_{n}^{(2)} )$ , where h n ( i ) ∈ l 1 ( N ) ∩ l 2 ( N ) $h_{n}^{(i)} \in l^{1} ( \mathbb{N} ) \cap l^{2} ( \mathbb{N} )$ , i = 1 , 2 $i = 1,2$ , and h 0 ( 1 ) ≠ 0 $h_{0}^{(1)} \ne 0$ . Upon determining the sets of eigenvalues and spectral singularities of L, we prove that, under certain conditions, L has a finite number of eigenvalues and spectral singularities with finite multiplicity. |
| first_indexed | 2024-12-13T05:15:31Z |
| format | Article |
| id | doaj.art-602ddc27e03b4ec5afe1c1b55d479f87 |
| institution | Directory Open Access Journal |
| issn | 1687-1847 |
| language | English |
| last_indexed | 2024-12-13T05:15:31Z |
| publishDate | 2020-08-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Advances in Difference Equations |
| spelling | doaj.art-602ddc27e03b4ec5afe1c1b55d479f872022-12-21T23:58:26ZengSpringerOpenAdvances in Difference Equations1687-18472020-08-01202011910.1186/s13662-020-02851-2The spectrum of discrete Dirac operator with a general boundary conditionNimet Coskun0Nihal Yokus1Department of Mathematics, Karamanoglu Mehmetbey UnivercityDepartment of Mathematics, Karamanoglu Mehmetbey UnivercityAbstract In this paper, we aim to investigate the spectrum of the nonselfadjoint operator L generated in the Hilbert space l 2 ( N , C 2 ) $l_{2}(\mathbb{N},\mathbb{C}^{2})$ by the discrete Dirac system { y n + 1 ( 2 ) − y n ( 2 ) + p n y n ( 1 ) = λ y n ( 1 ) , − y n ( 1 ) + y n − 1 ( 1 ) + q n y n ( 2 ) = λ y n ( 2 ) , n ∈ N , $$ \textstyle\begin{cases} y_{n+1}^{ (2 )} - y_{n}^{ (2 )} + p_{n} y_{n}^{ (1 )} =\lambda y_{n}^{ (1 )},\\ - y_{n}^{ (1 )} + y_{n-1}^{ (1 )} + q_{n} y_{n}^{ (2 )} =\lambda y_{n}^{ (2 )}, \end{cases}\displaystyle \quad n\in \mathbb{N}, $$ and the general boundary condition ∑ n = 0 ∞ h n y n = 0 , $$ \sum_{n = 0}^{\infty } h_{n}y_{n} = 0, $$ where λ is a spectral parameter, Δ is the forward difference operator, ( h n $h_{n}$ ) is a complex vector sequence such that h n = ( h n ( 1 ) , h n ( 2 ) ) $h_{n} = ( h_{n}^{(1)}, h_{n}^{(2)} )$ , where h n ( i ) ∈ l 1 ( N ) ∩ l 2 ( N ) $h_{n}^{(i)} \in l^{1} ( \mathbb{N} ) \cap l^{2} ( \mathbb{N} )$ , i = 1 , 2 $i = 1,2$ , and h 0 ( 1 ) ≠ 0 $h_{0}^{(1)} \ne 0$ . Upon determining the sets of eigenvalues and spectral singularities of L, we prove that, under certain conditions, L has a finite number of eigenvalues and spectral singularities with finite multiplicity.http://link.springer.com/article/10.1186/s13662-020-02851-2EigenparameterSpectral analysisEigenvaluesSpectral singularitiesDiscrete equationDirac equation |
| spellingShingle | Nimet Coskun Nihal Yokus The spectrum of discrete Dirac operator with a general boundary condition Advances in Difference Equations Eigenparameter Spectral analysis Eigenvalues Spectral singularities Discrete equation Dirac equation |
| title | The spectrum of discrete Dirac operator with a general boundary condition |
| title_full | The spectrum of discrete Dirac operator with a general boundary condition |
| title_fullStr | The spectrum of discrete Dirac operator with a general boundary condition |
| title_full_unstemmed | The spectrum of discrete Dirac operator with a general boundary condition |
| title_short | The spectrum of discrete Dirac operator with a general boundary condition |
| title_sort | spectrum of discrete dirac operator with a general boundary condition |
| topic | Eigenparameter Spectral analysis Eigenvalues Spectral singularities Discrete equation Dirac equation |
| url | http://link.springer.com/article/10.1186/s13662-020-02851-2 |
| work_keys_str_mv | AT nimetcoskun thespectrumofdiscretediracoperatorwithageneralboundarycondition AT nihalyokus thespectrumofdiscretediracoperatorwithageneralboundarycondition AT nimetcoskun spectrumofdiscretediracoperatorwithageneralboundarycondition AT nihalyokus spectrumofdiscretediracoperatorwithageneralboundarycondition |