Resonant scattering for tunable quantum walks on graphs with tails

TL;DR

The paper advances resonant scattering theory for discrete-time quantum walks on graphs with tails by reducing resonances to eigenvalues of a finite-rank internal-graph matrix and then applying matrix perturbation theory to obtain explicit asymptotics for resonant energies and the scattering matrix. It builds a robust framework using complex distortion to define resonances, a spectral-mapping theorem to connect interior and tail dynamics, and stationary-state constructions to define generalized eigenfunctions and the scattering matrix. The main contributions include a detailed perturbative analysis of tunable Grover walks, second-order reductions, and explicit formulas for resonance shifts and scattering data, supported by concrete examples on cycles and complete graphs. The results elucidate how interior graph geometry and boundary perturbations govern resonant transport, with implications for controlling quantum-walk-based scattering in networked systems.

Abstract

We study the resonant scattering for discrete time quantum walks on graphs with some tails. In our arguments, we reduce the study of resonances to the perturbation of eigenvalues of a finite rank matrix associated with the internal graph. Then we can apply Kato's perturbation theory of matrices, and the reduction process of generalized eigenspaces allows us to derive an explicit asymptotic expansion of the scattering matrix. As a consequence, we obtain the resonant scattering at resonant energies.

Figures (12)

• Figure 1: The generalized eigenfunction to $-\psi" +V\psi =\lambda \psi$, $\lambda >0$, with the double barrier potential satisfies the asymptotic behavior $\psi (x)\sim e^{i\sqrt{\lambda}x}+\rho (\lambda) e^{-i\sqrt{\lambda} x}$ as $x\to -\infty$ and $\psi (x)\sim \tau (\lambda) e^{i\sqrt{\lambda}x}$ as $x\to \infty$. It is well-known that $|\rho (\lambda)|^2 + |\tau (\lambda)|^2 =1$. Even though $\lambda < \sup_x V(x)$, the reflected wave vanishes for some $\lambda$. A similar phenomenon is also known for the potential $V \in C^{\infty} ({\bf R}^3 )$ of the shape resonance model $H(h)= -(1/2)h^2 \Delta +V$ on ${\bf R}^3$ with small parameter $h > 0$. If $V (x) \to 0$ rapidly as $|x| \to \infty$, $H(h)$ has no positive eigenvalue. However, there exist some resonances of $H(h)$ near the Dirichlet eigenvalues of $(-(1/2)h^2 \Delta +V)|_D$ in a bounded domain $\Omega^i$.
• Figure 2: An example of $\Gamma = (V,A)$ with four tails. The internal graph acts as a perturbation. The closed paths of the internal graph are analogues of the bounded classical trajectories for the Schrödinger operators. Note that some tails can have a common boundary vertex. In this case, we have $v_{2,0} = v_{4,0}$.
• Figure 3: The initial state $\varphi_0$ has its support only on incoming edges of tails.
• Figure 4: In view of the scattering theory, we consider a solution $\psi \in \ell^{\infty} (A)$ to $U\psi = e^{-i\lambda} \psi$ such that $\psi$ satisfies the situation of this figure. $\alpha^{\flat}$ is the vector of complex intensities of incoming waves, and $\alpha^{\sharp} (\lambda)$ is that of outgoing waves.
• Figure 5: Left: $C_4$, the cycle with four vertices. / Right: $K_4$, the complete graph with four vertices.

Theorems & Definitions (32)

• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 3.1
• Lemma 3.2
• Definition 3.3
• Proposition 3.4
• Lemma 3.5
• Proposition 3.6
• Proposition 4.1