Trajectories of a quadratic differential related to a quasi-exactly solvable sextic oscillator

TL;DR

This paper investigates the existence of solutions to a quadratic algebraic equation related to quadratic differentials, focusing on the critical trajectories and their connection to a quasi-exactly solvable sextic oscillator.

Contribution

It provides new insights into the critical graph structure of quadratic differentials associated with a specific algebraic equation in the context of quantum oscillators.

Findings

Existence of solutions as Cauchy transforms of signed measures.

Characterization of finite critical trajectories of the quadratic differential.

Connection between algebraic equations and the geometry of quadratic differentials.

Abstract

In this paper, we discuss the existence of solution (as Cauchy transform of a signed measure) of a particular algebraic quadratic equation of the form $\mathcal{C}^{2}\left( z\right) +r\left( z\right) \mathcal{C}\left( z\right) +s\left( z\right) =0.$ This problem remains to describe the critical graph of a related polynomial quadratic differential; in particular, we discuss the existence of finite critical trajectories of this quadratic differential.