Asymptotic Properties of Generalized Eigenfunctions for Multi-dimensional Quantum Walks

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.

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A Discontinuity of the Energy of Quantum Walk in Impurities

Higuchi

Komatsu²,

Konno

et al. 2021

Symmetry

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We consider the discrete-time quantum walk whose local dynamics is denoted by a common unitary matrix C at the perturbed region {0,1,⋯,M−1} and free at the other positions. We obtain the stationary state with a bounded initial state. The initial state is set so that the perturbed region receives the inflow ωn at time n(|ω|=1). From this expression, we compute the scattering on the surface of −1 and M and also compute the quantity how quantum walker accumulates in the perturbed region; namely, the energy of the quantum walk, in the long time limit. The frequency of the initial state of the influence to the energy is symmetric on the unit circle in the complex plain. We find a discontinuity of the energy with respect to the frequency of the inflow.

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Asymptotic Properties of Generalized Eigenfunctions for Multi-dimensional Quantum Walks

Cited by 3 publications

References 20 publications

Eigenvalues and threshold resonances of a two-dimensional split-step quantum walk with strong shift

Eigenvalues and threshold resonances of a two-dimensional split-step quantum walk with strong shift

Complex translation methods and its application to resonances for quantum walks

A Discontinuity of the Energy of Quantum Walk in Impurities

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