Exact WKB method for radial Schrödinger equation

We revisit exact WKB quantization for radial Schrödinger problems from the modern resurgence perspective, with emphasis on how ``physically meaningful’’ quantization paths should be chosen and interpreted. Using connection formulae at simple turning points and at regular singular points, we show that the nontrivial-cycle data give the spectrum. In particular, for the $3$-dimensional harmonic oscillator and the $3$-dimensional Coulomb potential, we explicitly compute a closed contour which starts at $+\infty$, bulges into the $r<0$ sector to encircle the origin, and returns to $+\infty$. Also we propose that the appropriate slice of the closed path provides a physical local basis at $r=0$, which is used by an origin-to-$\infty$ open path. Via the change of variables $r=e^x$ ($x\in(-\infty,\infty)$), the origin data are pushed to the boundary condition of convergence at $x\to-\infty$, which renders the equivalence between open-connection and closed-cycle quantization transparent. The Maslov contribution from the regular singularity is incorporated either as a small-circle monodromy which is justified in terms of renormalization group, or, equivalently, as a boundary phase; we also develop an optimized/variational perturbation theory on exact WKB. Our analysis clarifies, in radial settings, how mathematical monodromy data and physical boundary conditions dovetail, thereby addressing recent debates on path choices in resurgence-based quantization.

💡 Research Summary

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This paper revisits the exact WKB quantization of radial Schrödinger problems from the modern resurgence viewpoint, focusing on the physically meaningful choice of integration paths. The authors treat the three‑dimensional harmonic oscillator and the three‑dimensional Coulomb potential as benchmark integrable models. By employing the well‑established connection formulas for simple turning points (Airy‑type) and for regular singularities (double poles at the origin), they demonstrate that the non‑trivial cycle data—namely the Voros periods and the Stokes multipliers—completely determine the bound‑state spectrum.

A central theme is the equivalence between two seemingly different quantization prescriptions. The first is a closed contour that starts at (r=+\infty) on the real axis, dips into the complex (r<0) half‑plane, encircles the origin once, and returns to (+\infty). The second is an open path that begins in a punctured neighbourhood of the origin, uses a locally physical basis (e.g. the Frobenius solution (u(r)\sim r^{l+1})), and proceeds to infinity. By mapping the radial coordinate via (r=e^{x}) (so that (x\in(-\infty,\infty))), the singular behaviour at the origin is translated into a boundary condition at (x\to -\infty). In this picture the open‑path condition “the subdominant solution vanishes at infinity” becomes identical to the closed‑cycle quantization condition.

For the harmonic oscillator, the authors compute the Stokes graph, identify the two turning points at (a_{1,2}=\pm\sqrt{2E}), and construct the monodromy matrix that includes the small‑circle contribution around the origin. The resulting quantization condition reads
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