Quantitative observability for the Schrödinger equation with an anharmonic oscillator
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arXiv
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| Natura: | Preprint |
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2025
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| _version_ | 1866915088056713216 |
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| author | Huang, Shanlin Wang, Gengsheng Wang, Ming |
| author_facet | Huang, Shanlin Wang, Gengsheng Wang, Ming |
| contents | This paper studies the observability inequalities for the Schrödinger equation associated with an anharmonic oscillator $H=-\frac{\d^2}{\d x^2}+|x|$. We build up the observability inequality over an arbitrarily short time interval $(0,T)$, with an explicit expression for the observation constant $C_{obs}$ in terms of $T$, for some observable set that has a different geometric structure compared to those discussed in \cite{HWW}. We obtain the sufficient conditions and the necessary conditions for observable sets, respectively. We also present counterexamples to demonstrate that half-lines are not observable sets, highlighting a major difference in the geometric properties of observable sets compared to those of Schrödinger operators $H=-\frac{\d^2}{\d x^2}+|x|^{2m}$ with $m\ge 1$.
Our approach is based on the following ingredients: First, the use of an Ingham-type spectral inequality constructed in this paper; second, the adaptation of a quantitative unique compactness argument, inspired by the work of Bourgain-Burq-Zworski \cite{Bour13}; third, the application of the Szegö's limit theorem from the theory of Toeplitz matrices, which provides a new mathematical tool for proving counterexamples of observability inequalities. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_01258 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Quantitative observability for the Schrödinger equation with an anharmonic oscillator Huang, Shanlin Wang, Gengsheng Wang, Ming Analysis of PDEs Optimization and Control This paper studies the observability inequalities for the Schrödinger equation associated with an anharmonic oscillator $H=-\frac{\d^2}{\d x^2}+|x|$. We build up the observability inequality over an arbitrarily short time interval $(0,T)$, with an explicit expression for the observation constant $C_{obs}$ in terms of $T$, for some observable set that has a different geometric structure compared to those discussed in \cite{HWW}. We obtain the sufficient conditions and the necessary conditions for observable sets, respectively. We also present counterexamples to demonstrate that half-lines are not observable sets, highlighting a major difference in the geometric properties of observable sets compared to those of Schrödinger operators $H=-\frac{\d^2}{\d x^2}+|x|^{2m}$ with $m\ge 1$. Our approach is based on the following ingredients: First, the use of an Ingham-type spectral inequality constructed in this paper; second, the adaptation of a quantitative unique compactness argument, inspired by the work of Bourgain-Burq-Zworski \cite{Bour13}; third, the application of the Szegö's limit theorem from the theory of Toeplitz matrices, which provides a new mathematical tool for proving counterexamples of observability inequalities. |
| title | Quantitative observability for the Schrödinger equation with an anharmonic oscillator |
| topic | Analysis of PDEs Optimization and Control |
| url | https://arxiv.org/abs/2501.01258 |