The Borg-Marchenko uniqueness theorem for complex potentials
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866912367085879296 |
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| author | Pushnitski, Alexander Štampach, František |
| author_facet | Pushnitski, Alexander Štampach, František |
| contents | We introduce and study a new theoretical concept of \textit{spectral pair} for a Schrödinger operator $H$ in $L^2(\mathbb{R}_{+})$ with a bounded \textit{complex-valued} potential. The spectral pair consists of a scalar measure and a complex-valued function. We show that in many ways, the spectral pair generalises the classical spectral measure to the non-self-adjoint case. First, extending the classical Borg-Marchenko theorem, we prove a uniqueness result: the spectral pair uniquely determines the operator $H$. Second, we derive asymptotic formulas for the spectral pair in the spirit of the classical result of Marchenko. In the case of real-valued potentials, we relate the spectral pair to the spectral measure of $H$. Lastly, we provide formulas for the spectral pair at a~simple eigenvalue of~$|H|$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_03248 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Borg-Marchenko uniqueness theorem for complex potentials Pushnitski, Alexander Štampach, František Spectral Theory Functional Analysis 34L40, 37K15 We introduce and study a new theoretical concept of \textit{spectral pair} for a Schrödinger operator $H$ in $L^2(\mathbb{R}_{+})$ with a bounded \textit{complex-valued} potential. The spectral pair consists of a scalar measure and a complex-valued function. We show that in many ways, the spectral pair generalises the classical spectral measure to the non-self-adjoint case. First, extending the classical Borg-Marchenko theorem, we prove a uniqueness result: the spectral pair uniquely determines the operator $H$. Second, we derive asymptotic formulas for the spectral pair in the spirit of the classical result of Marchenko. In the case of real-valued potentials, we relate the spectral pair to the spectral measure of $H$. Lastly, we provide formulas for the spectral pair at a~simple eigenvalue of~$|H|$. |
| title | The Borg-Marchenko uniqueness theorem for complex potentials |
| topic | Spectral Theory Functional Analysis 34L40, 37K15 |
| url | https://arxiv.org/abs/2503.03248 |