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arXiv · 2402.06478

Topology of Stokes Complex Related to a Polynomial Quadratic Differential : Phase Transitions and Number of Short Trajectories

Gliia Braek·Mondher Chouikhi·Faouzi Thabet
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Abstract

In this paper, we give a full description of the critical graph of the quadratic differential $\varpi_{a,\theta}$ defined on the Riemann sphere $\widehat{% %TCIMACRO{\U{2102} }% %BeginExpansion \mathbb{C} %EndExpansion }$ by $\varpi_{a,\theta}=-e^{2i\theta}\left( z-a\right) \left( z^{2}-1\right) dz^{2},$ where $\theta\in% %TCIMACRO{\U{211d} }% %BeginExpansion \mathbb{R} %EndExpansion ,$ and $a\in% %TCIMACRO{\U{2102} }% %BeginExpansion \mathbb{C} %EndExpansion .$. We prove that the existence and the number of short trajectories of $\varpi_{a,\theta}$ 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 $\theta\in\left\{ 0,\pi/4\right\} .$ We investigate these classifications to study an inverse spectral problem related to the complex cubic oscillator for Schr\"{o}dinger equation.

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Gliia Braek, Mondher Chouikhi, Faouzi Thabet. 2024-02-09. Topology of Stokes Complex Related to a Polynomial Quadratic Differential : Phase Transitions and Number of Short Trajectories. https://arxiv.org/abs/2402.06478

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