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 $$\mathbb {C}$$-plane of some wave functions resulting from the WKB-techniques applied to a Schrodinger equation with quartic oscillator and having some boundary condition. As a 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 re-scaled solutions of the Schrodinger equation, to obtain a quadratic algebraic equation of the form $${\mathcal {C}}^{2}(z) +r(z){\mathcal {C}}(z)+s(z)=0$$, where r, s are also polynomials. As a second step, we discuss the existence of solutions (in the form of Cauchy transform of a signed measure) of this algebraic equation. It remains to describe the critical graph of a related quadratic differential $$-p(z)dz^{2}$$ where p(z) is a quartic polynomial. In particular, we discuss the existence (and their number) of finite critical trajectories of this quadratic differential.