Topology of Stokes Complex Related to a Polynomial Quadratic Differential : Phase Transitions and Number of Short Trajectories

2024/02/09 by Gliia Braek, Braek, Gliia, Mondher Chouikhi +3 · 1 citation
Computer Science · Physics and Astronomy · #30E25 (Primary) 34A26 #30F15 (Secondary) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Nonlinear Dynamics and Pattern Formation #Quantum Mechanics and Non-Hermitian Physics #Quantum chaos and dynamical systems

paper · pdf · doi:10.48550/arxiv.2402.06478

openalex publication_date 2024/02/09 · openalex created_date 2024/02/13 · openalex updated_date 2026/07/28

Abstract

In this paper, we give a full description of the critical graph of the quadratic differential \varpi_a,θ defined on the Riemann sphere \widehat% %TCIMACRO\U2102 % %BeginExpansion ℂ %EndExpansion by \varpi_a,θ=-e^2iθ( z-a) ( z²-1) dz², where θ∈% %TCIMACRO\U211d % %BeginExpansion ℝ %EndExpansion , and a∈% %TCIMACRO\U2102 % %BeginExpansion ℂ %EndExpansion .. We prove that the existence and the number of short trajectories of \varpi_a,θ depend on the location of a in certain curves defined on the complex plane as the level sets of some harmonic functions. More focus will be to the cases θ∈\ 0,π/4\ . We investigate these classifications to study an inverse spectral problem related to the complex cubic oscillator for Schrödinger equation.

Cited by

• Cubic Oscillator: Geometric Approach and Zeros of Eigenfunctions

Related

• Spectral networks for polynomial cubic differentials
• Quadratic differentials, half-plane structures, and harmonic maps to graphs
• Algebraic equation and quadratic differential related to generalized Bessel polynomials with varying parameters
• Trajectories of a quadratic differential related to a quasi-exactly solvable sextic oscillator
• The critical orbit structure of quadratic polynomials in ℤ_p