Numerical Study of Quantum Resonances in Chaotic Scattering

TL;DR

This work studies quantum resonances in chaotic scattering by numerical means, focusing on a two-degree-of-freedom Triple Gaussian model. It develops a complex-scaling, finite-rank discretization framework using a DVR basis and tensor-product structure to compute resonances, while introducing and estimating the fractal dimension D(K_E) of the classical trapped set. The main finding is evidence for a Weyl-like scaling N_res ~ $\hbar^{-(D(K_E)+1)/2}$ in a small energy window as $\hbar\to 0$, linking the fractal geometry of classical trapped trajectories to quantum resonance counts. The results, including tests on Double Gaussian scattering, support the conjectured scaling and demonstrate the feasibility of the numerical approach for chaotic scattering systems with fractal trapped sets.

Abstract

This paper presents numerical evidence that for quantum systems with chaotic classical dynamics, the number of scattering resonances near an energy $E$ scales like $\hbar^{-\frac{D(K_E)+1}{2}}$ as $\hbar\to{0}$. Here, $K_E$ denotes the subset of the classical energy surface $\{H=E\}$ which stays bounded for all time under the flow generated by the Hamiltonian $H$ and $D(K_E)$ denotes its fractal dimension. Since the number of bound states in a quantum system with $n$ degrees of freedom scales like $\hbar^{-n}$, this suggests that the quantity $\frac{D(K_E)+1}{2}$ represents the effective number of degrees of freedom in scattering problems.

Figures (44)

• Figure 1: Triple gaussian potential
• Figure 2: Illustration of complex scaling: The three lines indicate the location of the rotated continuous spectrum for different values of $\alpha$, while the box at the top of the figure is the region in which resonances are counted. Eigenvalues which belong to different values of $\alpha$ are marked with different styles of points. As explained later, only eigenvalues near the region of interest are computed. This results in a seemingly empty plot.
• Figure 3: A sinc function with $m=0$, $\Delta{x}=1$.
• Figure 4: Illustration of resonance program parameters in configuration space: The lower-left corner of the mesh is $(X_0,Y_0)$, while the upper right corner is $(X_1,Y_1)$. The mesh contains $N_x\times{N_y}$ grid points, and a basis function $\phi_{mn}$ is placed at each grid point. Stars mark the centers of the potentials, the circles have radius $2\sigma$ (with $\sigma=1/3$), and $R$ is set to $1.4$. Parameters for the classical computation are depicted in Figure \ref{['fig:dim-prog-params']}.
• Figure 5: A typical trajectory: Stars mark the potential centers. In this case, $R=1.4$ and $E=0.5$. The circles drawn in the figure have radius $1$, and the disjoint union of their cotangent bundles form the Poincaré section. Trajectories start on the circle centered at bump 0 (the bumps are, counterclockwise, 0, 1, and 2) with some given angle $\theta$ and angular momentum $p_\theta$. This trajectory generates the finite sequence $(\dot{0},1,2,0,2,\infty)$. (Symbolic sequences are discussed later in the paper.) An illustration of resonance computation is depicted in Figure \ref{['fig:res-prog-params']}. The dashed line is the time-reversed trajectory with the same initial conditions, generating the sequence $(\infty,2,0,2,\dot{0})$.