On spectral asymptotic of quasi-exactly solvable quartic

TL;DR

This paper investigates the asymptotic behavior and monodromy of the quasi-exactly solvable spectrum of a specific quartic potential, proposing conjectures on the shape of spectral branching points and zeros of special polynomials.

Contribution

It introduces a conjecture linking the asymptotic configuration of spectral branching points to zeros of Yablonskii-Vorob'ev polynomials and provides numerical analysis of spectral monodromy.

Findings

Conjecture on the asymptotic shape of branching points

Numerical evidence supporting the spectral monodromy analysis

Alternative description of the spectral configuration shape

Abstract

Motivated by the earlier results, we study theoretically and numerically the asymptotics and the monodromy of the quasi-exactly solvable part of the spectrum of the quasi-exactly solvable quartic introduced by C.~M.~Bender and S.~Boettcher. In particular, we formulate a conjecture on the coincidence of the asymptotic shape of the configuration of the branching points of the latter quartic with the asymptotic shape of zeros of the Yablonskii-Vorob'ev polynomials recently described and present its (conjectural) alternative description. Further we present a numerical study of the spectral monodromy for the os- cillator in question.