Non-Self-Adjoint Hill Operators whose Spectrum is a Real Interval
Fuente:
arXiv
Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2024
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866915510055075840 |
|---|---|
| author | Papanicolaou, Vassilis G. |
| author_facet | Papanicolaou, Vassilis G. |
| contents | Let $H = -d^2/dx^2 + q(x)$, $x \in \mathbb{R}$, where $q(x)$ is a periodic potential, and suppose that the spectrum $σ(H)$ of $H$ is the positive semi-axis $[0, \infty)$. In the case where $q(x)$ is real-valued (and locally square-integrable) a well-known result of G. Borg states that $q(x)$ must vanish almost everywhere. However, as it was first observed by M.G. Gasymov, there is an abundance of complex-valued potentials for which $σ(H) = [0, \infty)$.
In this article we conjecture a characterization of all complex-valued entire potentials whose spectrum is $[0, \infty)$. We also present an analog of Borg's result for complex potentials. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_10266 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Non-Self-Adjoint Hill Operators whose Spectrum is a Real Interval Papanicolaou, Vassilis G. Spectral Theory Classical Analysis and ODEs 34B30, 34L40, 47E05, 47A10 Let $H = -d^2/dx^2 + q(x)$, $x \in \mathbb{R}$, where $q(x)$ is a periodic potential, and suppose that the spectrum $σ(H)$ of $H$ is the positive semi-axis $[0, \infty)$. In the case where $q(x)$ is real-valued (and locally square-integrable) a well-known result of G. Borg states that $q(x)$ must vanish almost everywhere. However, as it was first observed by M.G. Gasymov, there is an abundance of complex-valued potentials for which $σ(H) = [0, \infty)$. In this article we conjecture a characterization of all complex-valued entire potentials whose spectrum is $[0, \infty)$. We also present an analog of Borg's result for complex potentials. |
| title | Non-Self-Adjoint Hill Operators whose Spectrum is a Real Interval |
| topic | Spectral Theory Classical Analysis and ODEs 34B30, 34L40, 47E05, 47A10 |
| url | https://arxiv.org/abs/2409.10266 |