Complex translation methods and its application to resonances for quantum walks

TL;DR

This paper investigates the properties of resonances in multi-dimensional quantum walks by analyzing eigenvalues of complex translated operators, revealing conditions for their existence or nonexistence in specific models.

Contribution

It introduces a framework for studying resonances in quantum walks using complex translation, providing new insights into their existence in different physical scenarios.

Findings

Resonances are characterized as eigenvalues of complex translated operators.

Existence or nonexistence of resonances depends on specific perturbations.

Results apply to models with elastic scattering and non-penetrable barriers.

Abstract

In this paper, some properties of resonances for multi-dimensional quantum walks are studied. Resonances for quantum walks are defined as eigenvalues of complex translated time evolution operators in the pseudo momentum space. For some typical cases, we show some results of existence or nonexistence of resonances. One is a perturbation of an elastic scattering of a quantum walk which is an analogue of classical mechanics. Another one is a shape resonance model which is a perturbation of a quantum walk with a non-penetrable barrier.