Cubic Oscillator: Geometric Approach and Zeros of Eigenfunctions

Abstract

In this paper, we give a geometric approach to the cubic oscillator with three distinct turning points based on the $\mathcal{D}╱\mathcal{S}\mathcal{G}$\emph{\ correspondence }introduced in \cite{Thabet+al}. The existence of quantization conditions, depending on extra data for the potential, is related to some particular critical graphs of the quadratic differential $\hat{I}{»}^2\left(z-a\right)\left(z^2-1\right)d{z}^2$ where $\hat{I}»$ is a non vanishing complex number, $a\in \mathbb{C}╲\left\{-1,1\right\}$. We investigate this geometric approach in two level: the first level is studying an inverse spectral problem related to cubic oscillator. The second level describes the zeros locations of eigenfunctions related to this oscillator. Our results may provide a geometric proof of some questions related to cubic potential case.