Explicit expression of scattering operator of some quantum walks on impurities

Abstract

In this paper, we consider the scattering theory for a one-dimensional quantum walk with impurities which make reflections and transmissions. We focus on an explicit expression of the scattering operator. Our construction of the formula is based on the counting paths of quantum walkers. The Fourier transform of the scattering operator gives an explicit formula of the scattering matrix which is deeply related with the resonant-tunneling for quantum walks.

Feynman Integral in Quantum Walk, Barrier-top Scattering and Hadamard Walk

“…A similar argument to [KKMS,Lemma 4.2] shows that the absolute value of each eigenvalue of E is less than 1, and the dimension of each eigenspace associated with a non-zero eigenvalue coincides with the multiplicity. We denote the maximum of the absolute value of the eigenvalues of E by M .…”