Resonances in finitely perturbed quantum walks, resonance expansion and generic simplicity

TL;DR

This paper introduces a framework for analyzing resonances in finitely perturbed quantum walks, linking resonance properties to quantum walk dynamics and proving generic simplicity of these resonances.

Contribution

It defines resonances via scattering matrix poles, develops a resonance expansion for quantum walk evolution, and proves the generic simplicity of resonances.

Findings

Resonances are characterized as poles of the scattering matrix in the lower half-plane.

The time evolution can be expressed through a resonance expansion involving Jordan chains.

Most resonances are simple, with multiple resonances being exceptional cases.

Abstract

We define resonances for finitely perturbed quantum walks as poles of the scattering matrix in the lower half plane. We show a resonance expansion which describes the time evolution in terms of resonances and corresponding Jordan chains. In particular, the decay rate of the survival probability is given by the imaginary part of resonances and the multiplicity. We prove generic simplicity of the resonances, although there are quantum walks with multiple resonances.