Spectral theory of Schrödinger operators with potentials that are measures supported on ${\Bbb N}$
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866911173877694464 |
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| author | Safronov, Oleg |
| author_facet | Safronov, Oleg |
| contents | We discuss spectral properties of the one-dimensional Schrödinger operator with a potential of the form $\sum V(n)δ(x-n)$. Our main result says that the absolutely continuous spectum of such an operator covers an interval $[α^2,β^2]$, if $V\in \ell^4$ and the Fourier series $\sum e^{2i kn}V(n)$ is a function of $k$ that is square integrable over $[α,β]$. We prove that this result is sharp by constructing examples of potentials $V\notin\ell^2$ for which the spectrum of the Schrödinger operator is singular. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_20320 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Spectral theory of Schrödinger operators with potentials that are measures supported on ${\Bbb N}$ Safronov, Oleg Mathematical Physics 34A37, 47A10 We discuss spectral properties of the one-dimensional Schrödinger operator with a potential of the form $\sum V(n)δ(x-n)$. Our main result says that the absolutely continuous spectum of such an operator covers an interval $[α^2,β^2]$, if $V\in \ell^4$ and the Fourier series $\sum e^{2i kn}V(n)$ is a function of $k$ that is square integrable over $[α,β]$. We prove that this result is sharp by constructing examples of potentials $V\notin\ell^2$ for which the spectrum of the Schrödinger operator is singular. |
| title | Spectral theory of Schrödinger operators with potentials that are measures supported on ${\Bbb N}$ |
| topic | Mathematical Physics 34A37, 47A10 |
| url | https://arxiv.org/abs/2509.20320 |