Unified exact WKB framework for resonance -- Zel'dovich/complex-scaling regularization and rigged Hilbert space

TL;DR

The paper introduces a unified exact WKB framework to treat quantum resonances non-perturbatively, unifying Zel'dovich regularization, complex scaling, and rigged Hilbert space within a single analytic structure. By applying the formalism to the inverted Rosen-Morse potential, it derives resonance energies and rigorous connection formulas through Stokes geometry and A-cycles, demonstrating the equivalence and complementarity of regularization schemes. Key contributions include explicit WKB constructions for generalized Riccati equations, a robust quantization condition via nonperturbative cycles, and the construction of a rigged Hilbert space $\mathcal{H}_\varepsilon \subseteq \Phi^\times$ that accommodates resonant states as $\varepsilon\to0$, linking non-Hermitian quantum mechanics with rigorous spectral theory. The proposed framework clarifies the analytic structure of resonances, offers non-perturbative accuracy for unstable quantum systems, and provides a foundation for future applications in open quantum systems and mathematical physics.

Abstract

We develop a unified framework for analyzing quantum mechanical resonances using the exact WKB method. The non-perturbative formulation based on the exact WKB method works for incorporating the Zel'dovich regularization, the complex scaling method, and the rigged Hilbert space. While previous studies have demonstrated the exact WKB analysis in bound state problems, our work extends its application to quasi-stationary states. By examining the inverted Rosen--Morse potential, we illustrate how the exact WKB analysis captures resonant phenomena in a rigorous manner. We explore the equivalence and complementarity of different well-established regularizations à la Zel'dovich and complex scaling within this framework. Also, we find the most essential regulator of functional analyticity and construct a modified Hilbert space of the exact WKB framework for resonance, which is called the rigged Hilbert space. This offers a deeper understanding of resonant states and their analytic structures. Our results provide a concrete demonstration of the non-perturbative accuracy of exact WKB methods in unstable quantum systems.

Figures (6)

• Figure 1: Stokes graph near a turning point. The solid and dashed curves are the associated Stokes curves on the first and second Riemann sheets, respectively. The blue wavy line is a branch cut. After crossing a branch cut once, the correspondence to the indices $+$ and $-$ in $M$ must be reversed.
• Figure 2: Schematic illustration of Stokes graph for the inverted Rosen--Morse potential, $V(x)=1/\cosh^2x$, via the Zel'dovich transformation. This geometry itself is identical to the original one Morikawa:2025grx. The left/right panels are devoted to $\mathop{\mathrm{Im}}\nolimits\hbar\gtrless0$ and $\mathop{\mathrm{Im}}\nolimits E\gtrless0$. The black points are the turning points and the solid curves are the associated Stokes curves. The dashed Stokes curves mean the periodicity of $\mathop{\mathrm{Im}}\nolimits x\in[-\pi/2,\pi/2]$ due to the cosine function on $\mathop{\mathrm{Re}}\nolimits x=0$. The blue points are double poles and the blue wavy lines are branch cuts.
• Figure 3: Schematic illustration of Stokes graph for the inverted Rosen--Morse potential. Same as Fig. \ref{['fig:stokes_graph']}, but $\mathop{\mathrm{Im}}\nolimits\hbar\gtrless0$ and $\mathop{\mathrm{Im}}\nolimits E\lessgtr0$. For simplicity, $\mathop{\mathrm{Im}}\nolimits E\sim 0$.
• Figure 4: Quantization condition with $\mathop{\mathrm{Im}}\nolimits\hbar>0$ and $\mathop{\mathrm{Im}}\nolimits E\lesssim0$ for resonance. The red path on the left panel is a simple option to carry out the analytical continuation, along which we can expect the normalizability of the exact WKB solution. The right panel shows actual connection manipulations of it. The red solid line is on the Riemann sheet depicted and the red dashed line is on another Riemann sheet after passing through the branch cut; the red dotted curve is the analytical continuation at infinity. The nontrivial cycle, $A$-cycle, is shown as the green loop.
• Figure 5: Illustration of computing Eq. \ref{['eq:zel_quantization']}. $\beta=1$. Although $f(x)$ diverges near $|\mathop{\mathrm{Im}}\nolimits x|\to\infty$, $f(-x)=-f(x)$ and hence $f(x)$ does not contribute to $e^{I_{i\infty}}$. Note that $I_n$, $I_{i\infty}$ and $I_\infty$ are finite. (If $f=0$ then $e^{I_\infty}$ is divergent.)