From three-body resonances to bound states in a continuum: pole trajectories

TL;DR

The paper addresses how three-body bound states in the continuum (BIC) arise in a mass-imbalanced, one-dimensional three-body system by following resonance poles as system parameters vary. The authors use the complex scaling method (CSM) and the Gaussian expansion method (GEM) to compute complex energies $E^{(3)}$ for a Gaussian interaction $V(r)= v_0 \exp[-\mu_g (r/r_0)^2]$. They show that BICs appear under variations of the interaction strength $v_0$, the interaction-range parameter $\mu_g$, and especially the mass ratio $\beta$, with $v_0$ and $\mu_g$ driving irregular trajectories and $\beta$ producing oscillatory trajectories with multiple BIC locations. The work clarifies the balance between kinematic structure and two-body details in stabilizing few-body resonances and provides a parametric roadmap for realizing BICs in low-dimensional quantum systems.

Abstract

We investigate the formation of three-body bound states in the continuum by tracing pole trajectories in the complex energy plane under variation of system parameters. Using a one-dimensional model of two identical bosons and a distinguishable particle interacting via Gaussian potentials, we systematically vary the interaction strength, interaction range, and mass ratio. Our results confirm the parametric nature of few-body bound states in a continuum (BIC) and extend this characterization to a broader set of system parameters. Specifically, we find that variations of both interaction parameters and the mass ratio can lead to the formation of at least one three-body BIC. However, the exact shape of trajectories differs, and for the mass ratio variation we find a more regular pattern with multiple BIC locations. These results suggest that the mechanism of few-body BIC formation is more sensitive to the kinematic structure of the problem than to the specific details of the two-body interaction.

Figures (3)

• Figure 1: Trajectories of the three-body resonance pole in the complex energy plane for variations of the interaction parameters $v_0$ (blue solid line) and $\mu_g$ (red dashed line). For the $v_0$ variation, the parameter ranges from $v_0 = -1.81$ (right) to $v_0 = -14.51$ (left), and we keep $\mu_g = 1$ and $\beta = 5$ fixed. For the $\mu_g$ variation, the parameter ranges from $\mu_g = 2.22$ (right) to $\mu_g = 0.3$ (left), while $v_0 = -4$ and $\beta = 5$ are fixed.
• Figure 2: Trajectory of the three-body resonance pole in the complex energy plane as the mass ratio $\beta$ is varied from $\beta = 1$ (right) to $\beta = 20$ (left), and $v_0 = -4$ and $\mu_g = 1$ are kept constant.
• Figure 3: Decay width $\Gamma = -2\text{Im}(E^{(3)})$ as a function of the interaction parameters (a) $v_0$ and (b) $\mu_g$, shown on a logarithmic scale. Both panels show a clear minimum in the width, indicating the presence of a bound state in the continuum (BIC) for specific parameter values.