2025/10/13 by Morikawa, Okuto, Ogawa, Shoya
#FOS: Physical sciences #Nuclear Theory (nucl-th) #Quantum Physics (quant-ph)
paper · doi:10.48550/arxiv.2510.11766
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 +∞, 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=ex (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.