2020/09/30 by Takashi Komatsu, Norio Konno, Hisashi Morioka +1 · 2 citations
Computer Science · Mathematics · Physics and Astronomy · #Eigenfunction #Eigenvalues and eigenvectors #Mathematics #Path integral formulation #Physics #Quantum #Quantum Computing Algorithms and Architecture #Quantum Information and Cryptography #Quantum algorithm #Quantum mechanics #Quantum walk #Quantum-Dot Cellular Automata #S-matrix #Scattering #Statistical physics #math-ph #math.MP
paper · pdf · doi:10.1142/s0129055x21500197
published as Reviews in Mathematical Physics Vol. 33 (2021) 2150019 · 21 pages, 1 figure
arxiv created 2021/03/21 · arxiv updated 2021/03/23 · openalex publication_date 2021/03/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
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 the 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 construction of the scattering matrix by counting paths of quantum walkers. We also mention some remarks on the tunneling effect.