Critical graph of a polynomial quadratic differential related to a Schr\"odinger equation with quartic potential

Abstract

In this paper, we study the weak asymptotic in the plane of some wave functions resulting from the WKB techniques applied to a Shrodinger equation with quartic oscillator and having some boundary condition. In first step, we make transformations of our problem to obtain a Heun equation satisfied by the polynomial part of the WKB wave functions .Especially , we investigate the properties of the Cauchy transform of the root counting measure of a re-scaled solutions of the Schrodinger equation, to obtain a quadratic algebraic equation of the form $\mathcal{C}^{2}\left( z\right) +r\left( z\right) \mathcal{C}\left( z\right) +s\left( z\right) =0$, where $r,s$ are also polynomials. In second step, we discuss the existence of solutions (as Cauchy transform of a signed measures) of this algebraic equation.This problem remains to describe the critical graph of a related 4-degree polynomial quadratic differential $-p\left( z\right) dz^{2}$. In particular, we discuss the existence(and their number) of finite critical trajectories of this quadratic differential.