Asymptotic Properties of Generalized Eigenfunctions for Multi-dimensional Quantum Walks

@article{Komatsu2021AsymptoticPO,
  title={Asymptotic Properties of Generalized Eigenfunctions for Multi-dimensional Quantum Walks},
  author={Takashi Komatsu and Norio Konno and Hisashi Morioka and Etsuo Segawa},
  journal={Annales Henri Poincar{\'e}},
  year={2021}
}
We construct a distorted Fourier transformation associated with the multi-dimensional quantum walk. In order to avoid the complication of notations, almost all of our arguments are restricted to two dimensional quantum walks (2DQWs) without loss of generality. The distorted Fourier transformation characterizes generalized eigenfunctions of the time evolution operator of the QW. The 2DQW which will be considered in this paper has an anisotropy due to the bias of the shift of the free QW. Then we… 
1 Citations

Figures from this paper

A Discontinuity of the Energy of Quantum Walk in Impurities
TLDR
This paper presents a meta-analyses of the model derived from a simulation of the response of the immune system to the presence of carbon dioxide in the atmosphere.

References

SHOWING 1-10 OF 32 REFERENCES
Generalized eigenfunctions and scattering matrices for position-dependent quantum walks
We study the spectral analysis and the scattering theory for time evolution operators of position-dependent quantum walks. Our main purpose of this paper is the construction of generalized
Quantum walks with an anisotropic coin I: spectral theory
We perform the spectral analysis of the evolution operator U of quantum walks with an anisotropic coin, which include one-defect models, two-phase quantum walks, and topological phase quantum walks
Quantum walks with an anisotropic coin II: scattering theory
We perform the scattering analysis of the evolution operator of quantum walks with an anisotropic coin, and we prove a weak limit theorem for their asymptotic velocity. The quantum walks that we
Scattering and inverse scattering for nonlinear quantum walks
We study large time behavior of quantum walks (QWs) with self-dependent (nonlinear) coin. In particular, we show scattering and derive the reproducing formula for inverse scattering in the weak
Spectral Properties of Schrödinger Operators on Perturbed Lattices
We study the spectral properties of Schrödinger operators on perturbed lattices. We shall prove the non-existence or the discreteness of embedded eigenvalues, the limiting absorption principle for
Inverse Problems, Trace Formulae for Discrete Schrödinger Operators
We study discrete Schrödinger operators with compactly supported potentials on Zd. Constructing spectral representations and representing S-matrices by the generalized eigenfunctions, we show that
Microlocal properties of scattering matrices
ABSTRACT We consider the scattering theory for a pair of operators H0 and H = H0 + V on L2(M, m), where M is a Riemannian manifold, H0 is a multiplication operator on M, and V is a pseudodifferential
Inverse Scattering for Schrödinger Operators on Perturbed Lattices
We study the inverse scattering for Schrödinger operators on locally perturbed periodic lattices. We show that the associated scattering matrix is equivalent to the Dirichlet-to-Neumann map for a
Commutator methods for unitary operators
We present an improved version of commutator methods for unitary operators under a weak regularity condition. Once applied to a unitary operator, the method typically leads to the absence of
Quantum walks on graphs and quantum scattering theory
We discuss a particular kind of quantum walk on a general graph. We affix two semi-infinite lines to a general finite graph, which we call tails. On the tails, the particle making the walk simply
...
1
2
3
4
...