On continuum and resonant spectra from exact WKB analysis

Abstract

Resonance phenomena are central to many quantum systems, where resonant states are typically characterized by pole singularities of the S-matrix. In this work, we employ the complex scaling method (CSM) in conjunction with exact WKB analysis to elucidate the geometric structure of scattering problems that encompass both bound and resonant states. By analyzing the continuum spectrum via the exact WKB framework, we derive the S-matrix for the inverted Rosen--Morse potential and reveal its underlying complex-geometric features. Furthermore, we reinterpret the Aguilar--Balslev--Combes theorem, the foundation of CSM, from a geometric perspective, and discuss the physical significance of the Siegert boundary condition within a rigorously defined modified Hilbert space. Our analysis bridges scattering cross-sections and spectral theory, offering new geometric insights into quantum resonance and scattering phenomena.

Figures (4)

• 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: Schematic diagram of the energy scale for Stokes graphs. For energies below the threshold ($E \leq 0$), bound states are present, with the Stokes curve with index $-$ extending towards $\pm \infty$ along the red path, ensuring the normalizability of the discrete spectrum. In the gray region, the appropriate connection formula applies, and branch cuts may be needed. Beyond the threshold, where the potential becomes flat at large $\mathop{\mathrm{Re}}\nolimits x$, the Stokes curve extends in the $\mathop{\mathrm{Im}}\nolimits x$ direction. With the correct sheet choice and the Stokes curve with index $-$, normalizability along the red path is guaranteed, revealing resonances with a discrete but complex spectrum. The choice of path corresponds to the Siegert boundary condition, where incoming waves are prohibited. Assuming an incoming wave $e^{ikx}$ from $\mathop{\mathrm{Re}}\nolimits x \to -\infty$, scattering states (transmitted and reflected waves) appear, corresponding to the continuum spectrum.
• Figure 3: Ratio of the transmission coefficients, $|T_{\text{Airy}}|/|T_{\text{exact}}|$. Every parameter excluding $E$ is set to unity. $k$ and $\Tilde\xi$ are written by $E$, and the $k$-dependence of the ratio is shown. $k=0$ and $\sqrt{2}$ correspond to $E=0$ and $E=U_0=1$, respectively.
• Figure 4: Integration contour in the complex $k$-plane of the Green function. The green contour indicates the original integration to prove the completeness \ref{['eq:complete_no-res']}. We find that in $\mathop{\mathrm{Im}}\nolimits k<0$ there exist resonant poles additionally. An integration along the magenta contour rotated by $2\theta$ via CSM concludes these residues as a physical state depending on $\theta$.