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arXiv · 2009.03498

Generalized eigenfunctions for quantum walks via path counting approach

Takashi Komatsu·Norio Konno·Hisashi Morioka·Etsuo Segawa
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Abstract

We consider the time-independent scattering theory for time evolution operators of one-dimensional two-state quantum walks. The scattering matrix associated with the position-dependent quantum walk naturally appears in the asymptotic behavior at spatial infinity of generalized eigenfunctions. The asymptotic behavior of generalized eigenfunctions is a consequence of an explicit expression of the Green function associated with the free quantum walk. When the position-dependent quantum walk is a finite rank perturbation of the free quantum walk, we derive a kind of combinatorial constructions of the scattering matrix by counting paths of quantum walkers. We also mention some remarks on the tunneling effect.

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Takashi Komatsu, Norio Konno, Hisashi Morioka, Etsuo Segawa. 2020-09-08. Generalized eigenfunctions for quantum walks via path counting approach. https://doi.org/10.1142/s0129055x21500197

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