Cubic Oscillator: Geometric Approach and Zeros of Eigenfunctions

TL;DR

This paper introduces a geometric framework for analyzing the cubic oscillator with three turning points, linking quantization conditions and eigenfunction zeros to quadratic differentials and critical graphs.

Contribution

It presents a novel geometric approach to the cubic oscillator, connecting spectral problems and eigenfunction zeros through quadratic differentials and critical graph analysis.

Findings

Quantization conditions depend on critical graphs of quadratic differentials.

Eigenfunction zeros are characterized by geometric structures related to the potential.

The approach offers potential proofs for questions in cubic potential spectral theory.

Abstract

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.