Resonant scattering for tunable quantum walks on graphs with tails

TL;DR

This paper analyzes resonant scattering in quantum walks on graphs with tails, using eigenvalue perturbation theory to derive explicit asymptotic expansions of the scattering matrix at resonant energies.

Contribution

It introduces a method to study resonances via eigenvalue perturbations and provides explicit asymptotic formulas for the scattering matrix in quantum walks on graphs.

Findings

Explicit asymptotic expansion of the scattering matrix at resonant energies

Reduction of resonance analysis to eigenvalue perturbation of finite matrices

Application of Kato's perturbation theory to quantum walk scattering

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.