Spectral mapping theorem of an abstract quantum walk

2015/06/22 by Yusuke Higuchi, Etsuo Segawa, Higuchi, Yusuke +3 · 1 citation
Mathematics · Physics and Astronomy · #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Spectral Theory (math.SP) #math-ph #math.MP #math.SP

paper · pdf · doi:10.48550/arxiv.1506.06457

arxiv created 2016/06/01 · arxiv updated 2016/06/02

Abstract

Given two Hilbert spaces, H and K, we introduce an abstract unitary operator U on H and its discriminant T on K induced by a coisometry from H to K and a unitary involution on H. In a particular case, these operators U and T become the evolution operator of the Szegedy walk on a graph, possibly infinite, and the transition probability operator thereon. We show the spectral mapping theorem between U and T via the Joukowsky transform. Using this result, we have completely detemined the spectrum of the Grover walk on the Sierpiński lattice, which is pure point and has a Cantor-like structure.

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