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Cubic Oscillator: Geometric Approach and Zeros of Eigenfunctions

Faouzi Thabet, Gliia Braek, Marwa Mansouri, Mondher Chouikhi

TL;DR

The paper develops a geometric framework for the cubic oscillator with three turning points by leveraging the D/SG correspondence for a parametric quadratic differential $-\lambda^{2}(z-1)(z+1)(z-a)\,dz^{2}$. It classifies eigenvalue problems via Stokes-graph mutations, identifies accumulation directions of the spectrum, and analyzes the asymptotic distribution of zeros of eigenfunctions along Stokes lines. Two main applications are given: constructing potentials with infinitely many real zeros and deriving explicit eigenvalue asymptotics in a related Sturm–Liouville problem, with zeros constrained to lie on transformed Stokes lines. The results provide a geometric path to understanding cubic potentials and suggest extensions to higher-degree polynomial potentials and questions of WKB summability and zero interlacing.

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 $λ^{2}\left(z-a\right) \left( z^{2}-1\right) dz^{2}$ where $λ$ 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.

Cubic Oscillator: Geometric Approach and Zeros of Eigenfunctions

TL;DR

The paper develops a geometric framework for the cubic oscillator with three turning points by leveraging the D/SG correspondence for a parametric quadratic differential . It classifies eigenvalue problems via Stokes-graph mutations, identifies accumulation directions of the spectrum, and analyzes the asymptotic distribution of zeros of eigenfunctions along Stokes lines. Two main applications are given: constructing potentials with infinitely many real zeros and deriving explicit eigenvalue asymptotics in a related Sturm–Liouville problem, with zeros constrained to lie on transformed Stokes lines. The results provide a geometric path to understanding cubic potentials and suggest extensions to higher-degree polynomial potentials and questions of WKB summability and zero interlacing.

Abstract

In this paper, we give a geometric approach to the cubic oscillator with three distinct turning points based on the \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 where is a non vanishing complex number, . 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.

Paper Structure

This paper contains 22 sections, 17 theorems, 70 equations, 19 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and notations
  3. Review of $\mathcal{D\diagup SG}$ correspondence
  4. A parametric Quadratic Differential
  5. Level Sets
  6. $\mathcal{D\diagup SG}$ correspondence
  7. Short review of complex WKB
  8. Canonical domains and asymptotic expansions
  9. Elementary basis
  10. Transition matrices
  11. Eigenvalue problems related to the cubic oscillator
  12. Admissible half planes
  13. Classification of Eigenvalue problem
  14. Zeros of Eigenfunctions
  15. The Stokes Graph is of Type B
  16. ...and 7 more sections

Key Result

Theorem 6

In each of the domains $\Omega _{1},...,\Omega _{n_{\theta }}$, the critical graph $\Gamma _{a,\theta }$ of the quadratic differential $\varpi _{a,\theta }$ has the same structure; it splits the Riemann sphere into five half-plane and two strip domains. Moreover, $\Gamma_{a,\theta }$: In particular, any change of a critical graph structure should pass by a critical graph with at least one short t

Figures (19)

  • Figure 7: Stokes graph of type A; $(a\notin\chi_{\theta})$
  • Figure 8: Stokes graph of type B; $(a\in\mathcal{S}_{1,\theta })$
  • Figure 9: Stokes graph of type BB; $(a\in\mathcal{S}_{1,\theta })$
  • Figure 10: Stokes graph of type Tree; $(a=t_{\theta}, \theta\in[\frac{\pi}{8},\frac{3\pi}{8}[)$
  • Figure 12: $D^1_{H',\varepsilon}$: The unshaded domain
  • ...and 14 more figures

Theorems & Definitions (48)

  • Remark 2
  • Remark 3
  • Definition 5
  • Theorem 6: Thabet+al
  • Corollary 7
  • proof
  • Proposition 8: Thabet+al
  • Remark 9
  • Remark 10
  • Definition 11
  • ...and 38 more