Cubic Oscillator: Geometric Approach and Zeros of Eigenfunctions

2025/11/03 by Faouzi Thabet, Gliia Braek, Thabet, Faouzi +5
Mathematics · Physics and Astronomy · #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Geometry and complex manifolds #Quantum Mechanics and Non-Hermitian Physics #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.2511.02050

openalex publication_date 2025/11/03 · openalex created_date 2025/11/06 · openalex updated_date 2026/07/28

Abstract

In this paper, we give a geometric approach to the cubic oscillator with three distinct turning points based on the D\diagup SG correspondence introduced in \citeThabet+al. The existence of quantization conditions, depending on extra data for the potential, is related to some particular critical graphs of the quadratic differential λ²(z-a) ( z²-1) dz² where λ is a non vanishing complex number, a∈ ℂ\diagdown \ -1,1\. 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.

Citations

• Topology of Stokes Complex Related to a Polynomial Quadratic Differential : Phase Transitions and Number of Short Trajectories
• Asymptotic integration theory for f'' + P(z)f = 0
• Critical graph of a polynomial quadratic differential related to a\n Schr "odinger equation with quartic potential
• Quadratic differentials as stability conditions
• A Note on EVgrafov-Fedoryuk's theory and quadratic differentials
• Complex Zeros of Eigenfunctions of 1D Schrödinger Operators
• Schrödinger type eigenvalue problems with polynomial potentials: Asymptotics of eigenvalues

Related

• The real spectrum of the imaginary cubic oscillator: An expository proof
• Topology of Stokes Complex Related to a Polynomial Quadratic Differential : Phase Transitions and Number of Short Trajectories
• Algebraic treatment of the Pais-Uhlenbeck oscillator and its PT-variant
• A Weakly Turbulent solution to the cubic Nonlinear Harmonic Oscillator on ℝ² perturbed by a real smooth potential decaying to zero at infinity
• Spectral networks for polynomial cubic differentials