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 C 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 C 2 (z) + r (z) C (z) + s (z) = 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 (z) dz 2 . In particular, we discuss the existence(and their number) of finite critical trajectories of this quadratic differential.2010 Mathematics subject classification: 30C15, 31A35, 34E05.