Cubic Oscillator: Geometric Approach and Zeros of Eigenfunctions

In this paper, we give a geometric approach to the cubic oscillator with three distinct turning points based on the $\mathcal{D\diagup SG}$\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 $\lambda ^{2}\left(z-a\right) \left( z^{2}-1\right) dz^{2}$ where $\lambda$ is a non vanishing complex number, $a\in \mathbb{C}\diagdown \left\{ -1,1\right\}$. 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.