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

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 ${\varpi }_{a,\theta }=-e^{2i\theta }(z-a)(z^2-1)d{z}^2,$ where $\theta \in$ and $a\in$. 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 $\theta \in \{0,\pi \mathrm{/}4\}\mathrm{.}$ We investigate these classifications to study an inverse spectral problem related to the complex cubic oscillator for Schrödinger equation.