Characterize the Stokes complex for general anharmonicity α

Determine a complete characterization of the Stokes complex of the quadratic differential V(x;E,ℓ) dx^2 associated with the reduced potential V(x;E,ℓ)=x^{2α}-E+((ℓ+1/2)^2)/x^2 for general α>0 and parameters (E,ℓ) with Re ℓ>−1/2. The characterization should fully describe the number, locations, and connectivity of turning points and all vertical trajectories (edges) to the boundary points 0 and ∞_{k+1/2}, thereby identifying all admissible lines and quadrilaterals across the relevant parameter regimes (including the cases with two positive real turning points).

Background

The paper develops a complex WKB framework and introduces the Stokes complex as a graph whose vertices are turning points of the reduced potential V(x;E,ℓ) and boundary points 0 and ∞_{k+1/2}, with edges given by vertical trajectories. For α=1 (harmonic oscillator) the authors give a complete description of the Stokes complex, but for general α they only establish partial results.

In the general case, they can prove admissibility of specific quadrilaterals relevant to the spectral problem and existence of a strictly admissible line connecting ∞1 and ∞{−1} when E exceeds a threshold, yet they explicitly state they cannot fully characterize the Stokes complex. A full characterization would systematize the global structure (turning points and their connections) across sectors and parameter ranges, which is central to applying the WKB method to global spectral questions.