Asymptotic Properties of Generalized Eigenfunctions for Multi-dimensional Quantum Walks

  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}},
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… 
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