Uniqueness problem for accretive Schrödinger operators with complex singular coefficients
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arXiv
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| Natura: | Preprint |
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2025
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| _version_ | 1866917121079902208 |
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| author | Mikhailets, Vladimir Molyboga, Volodymyr |
| author_facet | Mikhailets, Vladimir Molyboga, Volodymyr |
| contents | The paper studies the uniqueness problem for the one-dimensional Schrödinger operator associated with the formal differential expression \begin{equation*} l[u] =-u''+qu + i[(ru)'+ru'], \end{equation*} in the complex Hilbert space $L^{2}(\mathbb{R})$. The coefficients of the expression are complex-valued and satisfy \begin{equation*} q=s+Q', \quad s \in L^1_{loc}\left(\mathbb{R}\right) \quad\text{and}\quad Q, r \in L^2_{loc}\left(\mathbb{R}\right), \end{equation*} where the derivative is understood in the sense of distributions. In particular, the potential $q$ can be a Radon measure on the line. With the help of specially selected quasi-derivatives, the expression $l$ is treated as quasi-differential. The domains of the minimal $\mathrm{L}_{0}$ and maximal $\mathrm{L}$ operators associated with the expression $l$ in the space $L^{2}(\mathbb{R})$ are described. We find constructive conditions on the behaviour of $\mathrm{Im}\,r$ near $\pm \infty$ that guarantee that $\mathrm{L}_{0}=\mathrm{L}$ if the operator $\mathrm{L}_{0}$ is accretive. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_03215 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Uniqueness problem for accretive Schrödinger operators with complex singular coefficients Mikhailets, Vladimir Molyboga, Volodymyr Spectral Theory 34B20, 34B24, 34L40 The paper studies the uniqueness problem for the one-dimensional Schrödinger operator associated with the formal differential expression \begin{equation*} l[u] =-u''+qu + i[(ru)'+ru'], \end{equation*} in the complex Hilbert space $L^{2}(\mathbb{R})$. The coefficients of the expression are complex-valued and satisfy \begin{equation*} q=s+Q', \quad s \in L^1_{loc}\left(\mathbb{R}\right) \quad\text{and}\quad Q, r \in L^2_{loc}\left(\mathbb{R}\right), \end{equation*} where the derivative is understood in the sense of distributions. In particular, the potential $q$ can be a Radon measure on the line. With the help of specially selected quasi-derivatives, the expression $l$ is treated as quasi-differential. The domains of the minimal $\mathrm{L}_{0}$ and maximal $\mathrm{L}$ operators associated with the expression $l$ in the space $L^{2}(\mathbb{R})$ are described. We find constructive conditions on the behaviour of $\mathrm{Im}\,r$ near $\pm \infty$ that guarantee that $\mathrm{L}_{0}=\mathrm{L}$ if the operator $\mathrm{L}_{0}$ is accretive. |
| title | Uniqueness problem for accretive Schrödinger operators with complex singular coefficients |
| topic | Spectral Theory 34B20, 34B24, 34L40 |
| url | https://arxiv.org/abs/2512.03215 |