The Regularity of ODEs and Thimble Integrals with Respect to Borel Summation

TL;DR

The paper introduces a geometric framework for Borel summation, defining Borel regularity and showing that solutions to level-1 ODEs and one-dimensional thimble integrals are Borel regular. It develops a translation-surface viewpoint for Laplace and Borel transforms, converting irregular singularities into regular position-domain problems and enabling constructive proofs via integral equations. The authors prove existence, uniqueness, and Borel summability of key solutions, and illustrate the theory with Airy, Airy–Lucas, modified Bessel, generalized Airy, and a triangular cantilever example. The work illuminates resurgence phenomena and Stokes phenomena from the position-domain perspective, offering concrete, algorithmic routes from formal trans-monomial expansions to analytic solutions and their Borel sums.

Abstract

Through Borel summation, one can often reconstruct an analytic solution of a problem from its asymptotic expansion. We view the effectiveness of Borel summation as a regularity property of the solution, and we show that the solutions of certain differential equation and integration problems are regular in this sense. By taking a geometric perspective on the Laplace and Borel transforms, we also clarify why "Borel regular" solutions are associated with special points on the Borel plane. The particular classes of problems we look at are level 1 ODEs and exponential period integrals over one dimensional Lefschetz thimbles. To expand the variety of examples available in the literature, we treat various examples of these problems in detail.

Figures (12)

• Figure 1: The Laplace transform $\mathcal{L}^\theta_{\zeta, \alpha}$ integrates a function along the ray $\zeta \in \alpha + {\rm e}^{{\rm i}\theta}(0, \infty)$ in the position domain, turning it into a function on some half-plane $\mathop{\mathrm{Re}}\nolimits\bigl({\rm e}^{{\rm i}\theta} z\bigr) > \Lambda$ in the frequency domain.
• Figure 2: On the left, we see the rays $\mathcal{J}^{\pi/8}_{\zeta, \pm 1}$ on the translation surface $B = \mathbb{C}$, where $\zeta$ is the standard coordinate. On the right, we see the complex manifold $X = \mathbb{C}$, colored according to a holomorphic map $f \colon X \to B$ whose critical values are $\zeta = \pm 1$. The Lefschetz thimbles over the rays $\mathcal{J}^{\pi/8}_{\zeta, \pm 1}$ are shown.
• Figure 3: The frequency coordinate $z$ on the cotangent spaces of an ordinary point and a singular point. The singularity shown here has cone angle $6\pi$, like the singularities of the translation surface associated with the Airy function.
• Figure 4: The frequency coordinate $z$ on the cotangent spaces of an ordinary point and a singular point. The singularity shown here has cone angle $6\pi$, like the singularities of the translation surface associated with the Airy function.
• Figure 5: A sector $\Omega_\alpha$ in the position domain, and the corresponding union of half-planes $\widehat{\Omega}_\alpha^\Lambda$ in the frequency domain.

Theorems & Definitions (47)

• Definition 1.1
• Definition 1.2
• Definition 1.3
• Definition 1.4
• Theorem 1.5: Theorem \ref{['re:thm:exist_uniq_ODE']}
• Theorem 1.6: Theorem \ref{['re:thm:soln_is_Borel_sum']}
• Corollary 1.7: Corollary \ref{['re:cor:soln_borel-reg']}
• Definition 1.8
• Lemma 1.9: adapted from pham
• Theorem 1.10: Theorem \ref{['thm:maxim-proof']}