On the perturbed periodic Schrödinger operators with separate resonant embedded eigenvalues
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866918133771534336 |
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| author | Lyu, Kang Yang, Chuanfu |
| author_facet | Lyu, Kang Yang, Chuanfu |
| contents | In this paper, we consider Schrödinger operators on $L^2(0,\infty)$ given by
\begin{align}
Hu=(H_0+V)u=-u^{\prime\prime}+V_0u+Vu,\nonumber
\end{align}
where $V_0$ is real, $1$-periodic and $V$ is the perturbation. It is well known that under perturbations $V(x)=o(1)$ as $x\to\infty$, the essential spectrum of $H$ coincides with the essential spectrum of $H_0$. We introduce a new way to construct oscillatory decaying perturbations with resonant embedded eigenvalues. Given any at most countable set $S$ inside the essential spectrum, we can construct perturbations with $S$ contained in the set of eigenvalues if the resonant eigenvalues in $S$ satisfy some condition. In particular, if $S$ is a finite set (or countable set), we can construct perturbation with $V(x)=\frac{O(1)}{x}$ $\left(\mathrm{or}\ \abs{V(x)}\leq\frac{h(x)}{1+x}\right)$ as $x\to\infty$ if the resonant eigenvalues of $S$ appear in the same spectral bands or large separate spectral bands, where $h(x)$ is any given function with $\lim_{x\to\infty}h(x)=\infty$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_09509 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the perturbed periodic Schrödinger operators with separate resonant embedded eigenvalues Lyu, Kang Yang, Chuanfu Mathematical Physics Spectral Theory Primary: 34L05. Secondary: 34A30 In this paper, we consider Schrödinger operators on $L^2(0,\infty)$ given by \begin{align} Hu=(H_0+V)u=-u^{\prime\prime}+V_0u+Vu,\nonumber \end{align} where $V_0$ is real, $1$-periodic and $V$ is the perturbation. It is well known that under perturbations $V(x)=o(1)$ as $x\to\infty$, the essential spectrum of $H$ coincides with the essential spectrum of $H_0$. We introduce a new way to construct oscillatory decaying perturbations with resonant embedded eigenvalues. Given any at most countable set $S$ inside the essential spectrum, we can construct perturbations with $S$ contained in the set of eigenvalues if the resonant eigenvalues in $S$ satisfy some condition. In particular, if $S$ is a finite set (or countable set), we can construct perturbation with $V(x)=\frac{O(1)}{x}$ $\left(\mathrm{or}\ \abs{V(x)}\leq\frac{h(x)}{1+x}\right)$ as $x\to\infty$ if the resonant eigenvalues of $S$ appear in the same spectral bands or large separate spectral bands, where $h(x)$ is any given function with $\lim_{x\to\infty}h(x)=\infty$. |
| title | On the perturbed periodic Schrödinger operators with separate resonant embedded eigenvalues |
| topic | Mathematical Physics Spectral Theory Primary: 34L05. Secondary: 34A30 |
| url | https://arxiv.org/abs/2410.09509 |