118 Faouzi Thabet

Trajectories of Particular Quadratic Differentials on the Riemann Sphere // Abstract In this lecture, we give some basics of the theory of Quadratic Differentials on the Riemann Sphere. In the first part, the focus will be on the investigation of the existence of finite critical trajectories, and the description of the critical graph of some quadratic differentials related to solutions as Cauchy transform of a signed measure of an algebraic quadratic equation as the form : $p\left( z\right) \mathcal{C}^{2}\left( z\right) +q\left( z\right) \mathcal{C}\left( z\right) +r=0,$ for some polynomials $p,$ $q$ and $r.$ As an example, we study the large-degree analysis of the behaviour of the generalized Laguerre polynomials $L_{n}^{(\alpha )}$ when the parameters are complex and depend on the degree $n$ linearly. // In the second part, we describe the critical graph of a polynomial quadratic differential related to the Schr\"{o}dinger equation with cubic potential.