Critical graph of a polynomial quadratic differential related to a Schrödinger 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 $-p\left(z\right)d{z}^2$. In particular, we discuss the existence(and their number) of finite critical trajectories of this quadratic differential.

24 pages, 7 figures. arXiv admin note: text overlap with arXiv:1712.06090