Critical graph of a polynomial quadratic differential related to a Schrödinger equation with quartic potential


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 Schrödinger 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 rs 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.

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Part of this work was carried out during a visit of F.T to the university of Stockholm, Sweden. We would like to thank Professor Boris Shapiro for many helpful discussions. The authors acknowledge the contribution of the anonymous referee whose careful reading of the manuscript helped to improve the presentation.

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Correspondence to Faouzi Thabet.

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Chouikhi, M., Thabet, F. Critical graph of a polynomial quadratic differential related to a Schrödinger equation with quartic potential. Anal.Math.Phys. 9, 1905–1925 (2019).

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  • Quantum mechanics
  • WKB-analysis
  • Quadratic differentials
  • Cauchy transform

Mathematics Subject Classification

  • 30C15
  • 31A35
  • 34E05