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Geometry and Resurgence of WKB Solutions of Schrödinger Equations

Nikita Nikolaev

Abstract

We prove that formal WKB solutions of Schrödinger equations on Riemann surfaces are resurgent. Specifically, they are Borel summable in almost all directions and their Borel transforms admit endless analytic continuation away from a discrete subset of singularities. Our approach is purely geometric, relying on understanding the global geometry of complex flows of meromorphic vector fields using techniques from holomorphic Lie groupoids and the geometry of spectral curves. This framework provides a fully geometric description of the Borel plane, Borel singularities, and the Stokes rays. In doing so, we introduce a geometric perspective on resurgence theory.

Geometry and Resurgence of WKB Solutions of Schrödinger Equations

Abstract

We prove that formal WKB solutions of Schrödinger equations on Riemann surfaces are resurgent. Specifically, they are Borel summable in almost all directions and their Borel transforms admit endless analytic continuation away from a discrete subset of singularities. Our approach is purely geometric, relying on understanding the global geometry of complex flows of meromorphic vector fields using techniques from holomorphic Lie groupoids and the geometry of spectral curves. This framework provides a fully geometric description of the Borel plane, Borel singularities, and the Stokes rays. In doing so, we introduce a geometric perspective on resurgence theory.

Paper Structure

This paper contains 37 sections, 24 theorems, 95 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Main Results.
  3. Context.
  4. Generalised global flows.
  5. Narrative outline of the proof.
  6. Acknowledgements.
  7. Funding.
  8. Elements of Resurgence
  9. Baseline notation and terminology.
  10. Arcs and rays.
  11. Halflines and halfstrips.
  12. The Real-Oriented Blowup
  13. Tangent circle.
  14. Endless Riemann Surfaces
  15. Analytic continuation.
  16. ...and 22 more sections

Key Result

Proposition 1.1

Suppose $(X,\pi)$ is an endless Riemann surface and $p \in X$ is a regular point. Then the number of Stokes rays at $p$ is at most countable. Consequently, the union of all Stokes cuts is a closed subset of $\mathbb{C}$. Upon translating $\pi (p)$ to the origin for convenience, the complement of Sto

Figures (5)

  • Figure 1: All three paths $\gamma,\gamma',\gamma"$ pictured in (a) are critical paths on the punctured spectral curve $\sf{\Sigma}_\sf{\Delta}$: they start at $p$ and terminate at ramification points depicted by circled crosses, so they all determine distinct elements of $\Gamma_p$. The concatenated path $\gamma^{{\text{--}1}} \circ \gamma"$ is a loop in $\sf{\Sigma}_\sf{\Delta}$ based at $p$, hence an element of the fundamental group $\pi_1 (\sf{\Sigma}_\sf{\Delta}, p)$, yet $\gamma$ and $\gamma'$ are not related by any loop on $\sf{\Sigma}_\sf{\Delta}$. This exhibits the fact that $\Gamma_p$ is a disjoint union of several $\pi_1 (\sf{\Sigma}_\sf{\Delta}, p)$-torsors. The paths $\gamma$ and $\gamma'$ are critical trajectories because their projections to $\mathbb{C}$ via the central charge $\mathnormal{Z}_p$ pictured in (b) are straight line segments $[0,\xi]$ and $[0,\xi']$, so $\gamma, \gamma' \in \Gamma_p^0$. On the other hand, the path $\gamma"$ is not a critical trajectory because its projection $\wp"$ is not straight. Moreover, $\gamma' \notin \Gamma_p^0$ because $\wp"$ is not homotopic to the straight line segment $[0,\xi"]$ due to the obstructing singular point $\xi'$. The discrete subset $\Xi_p^0 = \mathnormal{Z}_p (\Gamma_p^0)$ is depicted in (d) by red crosses, and there are radial cuts emerging from each point. Their complement $\mathbb{E}_p$ contains infinite rays, such as $\mathbb{R}_\alpha \mathrel{\mathop:}= e^{i \alpha} \mathbb{R}_+$ and $\mathbb{R}_{\alpha'}$, which have a (locally) biholomorphic lift to infinitely long trajectories on the spectral curve $\sf{\Sigma}_\sf{\Delta}$ pictured in (c). In the depicted situation, these trajectories both fall into a pole in $\sf{\Delta}$. Notice that the path $\wp"$ from $0$ to $\xi"$ crosses the radial cut emerging from $\xi'$. Thus, the singularity represented by the path $\gamma"$ is not visible from the point $p$ in the sense that it cannot be reached along a trajectory, which is why the singular value at $\xi"$ (as well as many other singular values pictured in (b)) has been greyed out and there is no radial cut emerging from it. (Caption continues on the next page.)
  • Figure 1: (Caption continued from previous page.) The infinite rays that bypass the singularity $\xi$ slightly on the left and slightly on the right yield a pair of Borel resummed WKB solutions. To describe the discontinuous jump across the Stokes ray in the direction of $\xi$, we must investigate the presence of relevant saddle trajectories as depicted in (e). The corresponding critical trajectory $\gamma$ hits the ramification point $p_1$ which can be circumvented on the left or on the right. Continuing on the left, we follow the trajectory $\gamma_\rm{L}$ which is regular because it falls into a pole in $\sf{\Delta}$. On the other hand, if we continue on the right, we follow the trajectory $\gamma_\rm{R}$ which is critical because it hits another ramification point $p_2$. Thus, $\gamma_\rm{R}$ is a saddle trajectory, and the central charge of the concatenated path $\gamma_\rm{R} \circ \gamma$ is the singular value $\xi_1$ which lies on the radial cut emerging from $\xi$, as depicted in (f). Notice that this singular value was not visible when we chose to circumvent $\xi$ on the left, but it is visible if we do so on the right. In the depicted situation, these trajectories both fall into a pole in $\sf{\Delta}$. Upon reaching $p_2$, we must again make a choice of on what side to circumvent the singularity. If we choose the right side, we follow the trajectory $\gamma_{\rm{RR}}$ which we imagine is regular because it continues uninterrupted and, say, falls into another pole in $\sf{\Delta}$. Consequently, the trajectory $\gamma_\rm{RR}$ projects via $\mathnormal{Z}_p$ to an straight halfline line in $\mathbb{C}$ emanating from $\xi_1$. On the other hand, if choose the left side, we follow the trajectory $\gamma_{\rm{RL}}$ which is another saddle trajectory. The central charge of the concatenated path $\gamma_{\rm{RL}} \circ \gamma_{\rm{R}} \circ \gamma$ is the new singular value $\xi_2$ which also lies on the cut emerging from $\xi_1$. Again, this singularity is only visible if we make the correct sequence of turns at each singular encountered singularity (in this case, "first right, then left"), and it corresponds to a specific finite chain of composable saddle trajectories (in this case, $\gamma_{\rm{RL}} \circ \gamma_{\rm{R}}$). The saddle trajectory $\gamma_{\rm{RL}}$ hits a ramification point, and we can repeat the analysis. In this case, this ramification is actually again the point ramification point $p_1$; i.e., the path $\gamma_{\rm{RL}} \circ \gamma_{\rm{R}}$ is in fact a homology cycle of $\sf{\Sigma}_\sf{\Sigma}$. So, in a situation like this, we can add an arbitrary multiple of this homology cycle and therefore discover countably-infinitely many new singular values located on the cut emerging from $\xi$, each a constant integer multiple translate of another.
  • Figure 2: The real-oriented blowup of $X$ at $p$ and a sector at $p$ with opening $A = (\alpha_1, \alpha_2)$.
  • Figure 3: Visible singularities, Stokes cuts, and Stokes diagrams.
  • Figure 4: Stability of regular rays. Stability of Stokes rays is similar.

Theorems & Definitions (61)

  • Definition 1.1: real-oriented blowup
  • Definition 1.2: sectors
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5: isolated singularities
  • Definition 1.6: endless analytic continuation
  • Definition 1.7
  • Definition 1.8: asymptotic and infinite paths
  • Definition 1.9: geodesics
  • Definition 1.10: visible singularities; regular and Stokes rays
  • ...and 51 more