Cubic Oscillator: Geometric Approach and Zeros of Eigenfunctions
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918203035222016 |
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| author | Thabet, Faouzi Braek, Gliia Mansouri, Marwa Chouikhi, Mondher |
| author_facet | Thabet, Faouzi Braek, Gliia Mansouri, Marwa Chouikhi, Mondher |
| contents | In this paper, we give a geometric approach to the cubic oscillator with three distinct turning points based on the $\mathcal{D\diagup SG}$\emph{\ correspondence }introduced in \cite{Thabet+al}. The existence of quantization conditions, depending on extra data for the potential, is related to some particular critical graphs of the quadratic differential $λ^{2}\left(z-a\right) \left( z^{2}-1\right) dz^{2}$ where $λ$ is a non vanishing complex number, $a\in \mathbb{C}\diagdown \left\{ -1,1\right\}$. We investigate this geometric approach in two level: the first level is studying an inverse spectral problem related to cubic oscillator. The second level describes the zeros locations of eigenfunctions related to this oscillator. Our results may provide a geometric proof of some questions related to cubic potential case. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2511_02050 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Cubic Oscillator: Geometric Approach and Zeros of Eigenfunctions Thabet, Faouzi Braek, Gliia Mansouri, Marwa Chouikhi, Mondher Classical Analysis and ODEs In this paper, we give a geometric approach to the cubic oscillator with three distinct turning points based on the $\mathcal{D\diagup SG}$\emph{\ correspondence }introduced in \cite{Thabet+al}. The existence of quantization conditions, depending on extra data for the potential, is related to some particular critical graphs of the quadratic differential $λ^{2}\left(z-a\right) \left( z^{2}-1\right) dz^{2}$ where $λ$ is a non vanishing complex number, $a\in \mathbb{C}\diagdown \left\{ -1,1\right\}$. We investigate this geometric approach in two level: the first level is studying an inverse spectral problem related to cubic oscillator. The second level describes the zeros locations of eigenfunctions related to this oscillator. Our results may provide a geometric proof of some questions related to cubic potential case. |
| title | Cubic Oscillator: Geometric Approach and Zeros of Eigenfunctions |
| topic | Classical Analysis and ODEs |
| url | https://arxiv.org/abs/2511.02050 |