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.

Figures (3)

• Figure 1: Stokes graph for the $3$D harmonic oscillator. $\mathop{\mathrm{Re}}\nolimits E>0$. The black solid curves are the Stokes curves, and the black points are the turning points connecting three Stokes curves. The blue wavy line is a branch cut. The red arrowed curve is a possible physical path on which the wave function is defined. That is, this path means the normalizable wave function, which should be supposed to be subdominant along $r\to\infty\pm i\epsilon$ with small $\epsilon>0$.
• Figure 2: The Stokes graph with the regular singularity near the origin. The white circle is the origin and the regular singularity. The naive connection, $N_{a_1a_2}$, misses this nontrivial monodromy phase. We should improve it by adding the closed-loop integral around $r=0$.
• Figure 3: Stokes graph for the Coulomb potential. $\mathop{\mathrm{Re}}\nolimits E<0$. The solid curves correspond to the Stokes curves. The bullet denotes the turning point, $a$, and the open circle is the origin, $r=0$. The blue wavy line between $0$ and $a$ is the branch cut. The red path means the normalizable wave function, which should be supposed to be subdominant along $r\to\infty\pm i\epsilon$ with small $\epsilon>0$.

Theorems & Definitions (3)

• Proposition
• Theorem
• Theorem