Exact quantization method for the polynomial 1D Schrödinger equation | Méthode de quantification exacte pour l'équation de Schrödinger polynômiale 1D

André Voros (1)

1 SPhT - Service de Physique Théorique

The stationary 1D Schrödinger equation with a polynomial potential $V(q)$ of degree $N$ is reduced analytically to a system of (complex) exact quantization conditions of Bohr--Sommerfeld form. These reduced equations can be probed numerically for effective solvability: in test cases, they appear to generate discrete dynamical systems with contractive behavior, and converging towards the exact quantum data as their fixed point. (To date, this statement is verified only empirically and for potentials close to purely quartic or sextic ones.)

Mots clés: spectral determinant; Wronskian identity; Schrödinger equation; exact quantization; anharmonic oscillator

Domaines: Physique Générale [physics.gen-ph]

André Voros. Exact quantization method for the polynomial 1D Schrödinger equation | Méthode de quantification exacte pour l'équation de Schrödinger polynômiale 1D. Toward the Exact WKB Analysis of Differential Equations, Linear or Non-Linear. Conference on Algebraic analysis of singular perturbations, 2000, Kyoto, pp.97-108. ⟨hal-00087757⟩