2017/12/17 by Mondher Chouikhi, Chouikhi, Mondher, Faouzi Thabet +1 · 1 citation
Mathematics · Physics and Astronomy · #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Quantum Mechanics and Non-Hermitian Physics #Quantum chaos and dynamical systems #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.1712.06133
openalex publication_date 2017/12/17 · openalex created_date 2022/10/04 · openalex updated_date 2026/07/28
In this paper, we study the weak asymptotic in the plane of some wave\nfunctions resulting from the WKB techniques applied to a Shrodinger equation\nwith quartic oscillator and having some boundary condition. In first step, we\nmake transformations of our problem to obtain a Heun equation satisfied by the\npolynomial part of the WKB wave functions .Especially , we investigate the\nproperties of the Cauchy transform of the root counting measure of a re-scaled\nsolutions of the Schrodinger equation, to obtain a quadratic algebraic equation\nof the form \C2\( z\) +r\( z\) \C\(\nz\) +s\( z\) =0, where r,s are also polynomials. In second\nstep, we discuss the existence of solutions (as Cauchy transform of a signed\nmeasures) of this algebraic equation.This problem remains to describe the\ncritical graph of a related 4-degree polynomial quadratic differential\n-p\( z\) dz2. In particular, we discuss the existence(and their\nnumber) of finite critical trajectories of this quadratic differential.\n