Nonperturbative Formulation of Resonances in Quantum Mechanics Based on Exact WKB Method

Abstract

We study quasi-stationary states in quantum mechanics using the exact Wentzel--Kramers--Brillouin (WKB) analysis as a nonperturbative framework. Whereas previous works focused mainly on stable systems, we explore unstable states such as resonances. As a concrete example, we analyze the inverted Rosen--Morse potential, which exhibits barrier resonance. This model allows exact solutions, enabling a direct comparison with exact WKB predictions. We provide a simple analytic picture of resonance and demonstrate consistency between exact and WKB-based results, extending the applicability of exact WKB analysis to nonpolynomial potentials.

Figures (6)

• Figure 1: Distribution of S-matrix poles in (a) the complex $k$-plane and (b) the complex $E$-plane. In the right panel (b), the bound states exist in the first Riemann surface, while the resonant states appear in the second Riemann surface.
• Figure 2: Stokes graphs for the cubic potential. The black solid curves are the Stokes curves and black points are the turning points connecting three Stokes curves. The blue wavy lines are branch cuts. The red arrowed curve is a possible physical path on which the wave function is defined.
• Figure 3: Potential structure of harmonic oscillator, $-z^2$, in complex plane
• Figure 4: Potential structure of $1/\cosh^2{z}$ in complex plane
• Figure 5: Schematically illustrated Stokes graph for the inverted Rosen--Morse potential, $V(x)=1/\cosh^2x$. (Left)$\mathop{\mathrm{Im}}\nolimits\hbar>0$, (right)$\mathop{\mathrm{Im}}\nolimits\hbar<0$. The black points denote the turning points and the solid curves are the corresponding Stokes curves. The dashed ones, also 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 point is the double pole and the blue wavy lines are branch cuts.