Inverse problems associated with the Hill operator
Let $\ell_n$ be the length of the $n$-th instability interval of the Hill operator $Ly=-y''+q(x)y$. We prove that if $\ell_n=o(n^{-2})$ and the set $\{(n\pi)^2: n \text{ is even and } n>n_0\}$ is a subset of the periodic spectrum of the Hill operator, then $q=0$ a.e., where $n_0$ i...
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| Format: | Article |
| Language: | English |
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Texas State University
2016-01-01
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| Series: | Electronic Journal of Differential Equations |
| Subjects: | |
| Online Access: | http://ejde.math.txstate.edu/Volumes/2016/41/abstr.html |
| _version_ | 1818501825279557632 |
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| author | Alp Arslan Kirac |
| author_facet | Alp Arslan Kirac |
| author_sort | Alp Arslan Kirac |
| collection | DOAJ |
| description | Let $\ell_n$ be the length of the $n$-th instability interval of the Hill
operator $Ly=-y''+q(x)y$. We prove that if $\ell_n=o(n^{-2})$ and the set
$\{(n\pi)^2: n \text{ is even and } n>n_0\}$ is a subset of the periodic
spectrum of the Hill operator, then $q=0$ a.e., where $n_0$ is a sufficiently
large positive integer such that $\ell_n<\varepsilon n^{-2}$ for all
$n>n_0(\varepsilon)$ with some $\varepsilon>0$.
A similar result holds for the anti-periodic case. |
| first_indexed | 2024-12-10T21:01:37Z |
| format | Article |
| id | doaj.art-4ed45289895a4c07973c4f85b9260d91 |
| institution | Directory Open Access Journal |
| issn | 1072-6691 |
| language | English |
| last_indexed | 2024-12-10T21:01:37Z |
| publishDate | 2016-01-01 |
| publisher | Texas State University |
| record_format | Article |
| series | Electronic Journal of Differential Equations |
| spelling | doaj.art-4ed45289895a4c07973c4f85b9260d912022-12-22T01:33:47ZengTexas State UniversityElectronic Journal of Differential Equations1072-66912016-01-01201641,112Inverse problems associated with the Hill operatorAlp Arslan Kirac0 Pamukkale Univ., Denizli, Turkey Let $\ell_n$ be the length of the $n$-th instability interval of the Hill operator $Ly=-y''+q(x)y$. We prove that if $\ell_n=o(n^{-2})$ and the set $\{(n\pi)^2: n \text{ is even and } n>n_0\}$ is a subset of the periodic spectrum of the Hill operator, then $q=0$ a.e., where $n_0$ is a sufficiently large positive integer such that $\ell_n<\varepsilon n^{-2}$ for all $n>n_0(\varepsilon)$ with some $\varepsilon>0$. A similar result holds for the anti-periodic case.http://ejde.math.txstate.edu/Volumes/2016/41/abstr.htmlHill operatorinverse spectral theoryeigenvalue asymptoticsFourier coefficients |
| spellingShingle | Alp Arslan Kirac Inverse problems associated with the Hill operator Electronic Journal of Differential Equations Hill operator inverse spectral theory eigenvalue asymptotics Fourier coefficients |
| title | Inverse problems associated with the Hill operator |
| title_full | Inverse problems associated with the Hill operator |
| title_fullStr | Inverse problems associated with the Hill operator |
| title_full_unstemmed | Inverse problems associated with the Hill operator |
| title_short | Inverse problems associated with the Hill operator |
| title_sort | inverse problems associated with the hill operator |
| topic | Hill operator inverse spectral theory eigenvalue asymptotics Fourier coefficients |
| url | http://ejde.math.txstate.edu/Volumes/2016/41/abstr.html |
| work_keys_str_mv | AT alparslankirac inverseproblemsassociatedwiththehilloperator |