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 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.

arXiv comment30 pages, 8 figures. v2: Sections 5 (Langer transformation) and 6 (renormalization group and optimized/variational perturbation theory) are added. v3: added several appendices, including applications to nontrivial (non-solvable) examples, and a general statement on the equivalence between open-path and closed-cycle (sketch of proof). v4: to appear in J. Phys. A: Math. Theor

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