Resonant scattering for tunable quantum walks on graphs with tails

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.

💡 Research Summary

The paper investigates resonant scattering phenomena in discrete‑time quantum walks (DTQWs) defined on infinite graphs that consist of a finite internal subgraph together with several semi‑infinite “tails”. The authors first formalize the quantum walk as a unitary evolution operator (U = SC), where (S) is a shift (moving the walker along directed edges) and (C) is a coin operator that acts locally at each vertex. On the tails the coin is fixed to the simple swap matrix, guaranteeing perfect transmission in the unperturbed case.

A “free” walk (U_{0}=eU_{0}\oplus U_{\text{int}}) is introduced: (eU_{0}) acts as a pure shift on each tail, while (U_{\text{int}}) is a finite‑dimensional unitary matrix governing the internal graph. The actual walk (U) differs from (U_{0}) by a finite‑rank perturbation (W = U-U_{0}). Consequently the essential spectrum of both operators coincides with the whole unit circle (S^{1}), while any eigenvalues of (U) are confined to the internal subspace and are at most (#A_{\text{int}}) in number.

To define resonances the authors employ a complex distortion technique. For a complex parameter (\theta) they construct a unitary operator (Q(\theta)) that multiplies tail amplitudes by (e^{i\theta x}). The distorted walk (U(\theta)=Q(\theta)UQ(\theta)^{-1}) has essential spectrum shifted by the factor (e^{\operatorname{Im}\theta}). Resonances are identified as poles of the analytically continued resolvent ((U(\theta)-e^{-iz})^{-1}) that cross the real axis (or the unit circle) as (\theta) moves into the complex half‑plane.

The key technical step is the reduction of the resonance problem to a finite‑dimensional eigenvalue problem for a matrix (M(\lambda)) that encodes the coupling between the internal graph and the tails. The matrix depends analytically on the spectral parameter (\lambda) and on a tunable coupling constant (\varepsilon) that controls the openness of the “gates” at the boundary vertices. By applying Kato’s perturbation theory for matrices, together with a reduction process for generalized eigenspaces, the authors obtain explicit asymptotic expansions of the eigenvalues (\lambda_{r}(\varepsilon)) (the resonant energies) and of the associated eigenprojections.

These expansions feed directly into the scattering matrix. Using the projection (P) that maps tail states to the internal subspace, the scattering matrix is expressed as
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