Exact WKB method for radial Schrödinger equation

Abstract 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 (3D) harmonic oscillator and the 3D Coulomb potential, we explicitly compute a closed contour which starts at + ∞ , bulges into the r < 0 sector to encircle the origin, and returns to + ∞ . 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- ∞ open path. Via the change of variables r = e x ( x ∈ ( − ∞ , ∞ ) ), the origin data are pushed to the boundary condition of convergence at x → − ∞ , 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.