Exact WKB conjecture for higher-order WKB-regular connections with strongly GMN spectral curves
Establish the following exact WKB properties for an ℏ-flat connection ∇ that is WKB-regular and whose spectral curve is strongly GMN at an unobstructed angle θ: (i) show that the Borel–Laplace transform ℒΨ of any formal WKB solution Ψ is analytically continuable along ℝ_{>0}·e^{2πiθ and is Laplace-transformable in the direction θ when z ∈ C\D is not on the θ-Stokes graph; and (ii) for z ∈ C\D, characterize that the first sheet of the analytic continuation of ℒΨ is smooth on ℝ_{>0}·e^{2πiθ minus the discrete set {m(𝒯_i, θ)·e^{2πiθ}} determined by the masses of the Stokes trees 𝒯_i passing through z.
References
Exact WKB analysis has been expected for higher order differential equations. However, we cannot find much exact conjectures on exact WKB analysis in the literature. Here, as an application of our results, we give a version of an exact conjecture. Assume $\nabla$ is WKB-regular and the spectral curve is strongly GMN. Let $\theta$ be an unobstructed angle whose existence is assured by Theorem~\ref{thm:main}. 1. If $z\in C\bs D$ is not on the Stokes graph at $\theta$, $\cL\Psi$ is analytically continuable on $\bR_{>0}\cdot e{2\pi i\theta$. Moreover, $\cL\Psi$ is Laplace transformable in the direction $\theta$. 2. Let $\cT_1,..,\cT_i,...$ be the set of Stokes trees at $\theta$ passing through $z$. Then the first sheet of the analytic continuation of $\cL\Psi$ is smooth on $\bR_{>0}\cdot e{2\pi i\theta}\bs \lc m(\cT_i, \theta)\cdot e{2\pi i\theta}\rc_i.