Topology of Stokes Complex Related to a Polynomial Quadratic Differential : Phase Transitions and Number of Short Trajectories
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| Natura: | Preprint |
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2024
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| author | Braek, Gliia Chouikhi, Mondher Thabet, Faouzi |
| author_facet | Braek, Gliia Chouikhi, Mondher Thabet, Faouzi |
| contents | In this paper, we give a full description of the critical graph of the quadratic differential $\varpi_{a,θ}$ defined on the Riemann sphere $\widehat{% %TCIMACRO{\U{2102} }% %BeginExpansion \mathbb{C} %EndExpansion }$ by $\varpi_{a,θ}=-e^{2iθ}\left( z-a\right) \left( z^{2}-1\right) dz^{2},$ where $θ\in% %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ,$ and $a\in% %TCIMACRO{\U{2102} }% %BeginExpansion \mathbb{C} %EndExpansion .$. We prove that the existence and the number of short trajectories of $\varpi_{a,θ}$ depend on the location of $a$ in certain curves defined on the complex plane as the level sets of some harmonic functions. More focus will be to the cases $θ\in\left\{ 0,π/4\right\} .$ We investigate these classifications to study an inverse spectral problem related to the complex cubic oscillator for Schrödinger equation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_06478 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Topology of Stokes Complex Related to a Polynomial Quadratic Differential : Phase Transitions and Number of Short Trajectories Braek, Gliia Chouikhi, Mondher Thabet, Faouzi Classical Analysis and ODEs 30E25 (Primary) 34A26, 30F15 (Secondary) In this paper, we give a full description of the critical graph of the quadratic differential $\varpi_{a,θ}$ defined on the Riemann sphere $\widehat{% %TCIMACRO{\U{2102} }% %BeginExpansion \mathbb{C} %EndExpansion }$ by $\varpi_{a,θ}=-e^{2iθ}\left( z-a\right) \left( z^{2}-1\right) dz^{2},$ where $θ\in% %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ,$ and $a\in% %TCIMACRO{\U{2102} }% %BeginExpansion \mathbb{C} %EndExpansion .$. We prove that the existence and the number of short trajectories of $\varpi_{a,θ}$ depend on the location of $a$ in certain curves defined on the complex plane as the level sets of some harmonic functions. More focus will be to the cases $θ\in\left\{ 0,π/4\right\} .$ We investigate these classifications to study an inverse spectral problem related to the complex cubic oscillator for Schrödinger equation. |
| title | Topology of Stokes Complex Related to a Polynomial Quadratic Differential : Phase Transitions and Number of Short Trajectories |
| topic | Classical Analysis and ODEs 30E25 (Primary) 34A26, 30F15 (Secondary) |
| url | https://arxiv.org/abs/2402.06478 |