Spectral mapping theorem of an abstract quantum walk

DOI10.1007/S11128-019-2448-6zbMATH Open1508.05149arXiv1506.06457MaRDI QIDQ6262880

Etsuo Segawa, Akito Suzuki

Publication date: 21 June 2015

Abstract: Given two Hilbert spaces, $mathcalH$ and $mathcalK$, we introduce an abstract unitary operator $U$ on $mathcalH$ and its discriminant $T$ on $mathcalK$ induced by a coisometry from $mathcalH$ to $mathcalK$ and a unitary involution on $mathcalH$. 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 Sierpi'nski lattice, which is pure point and has a Cantor-like structure.

Mathematics Subject Classification ID

Graphs and linear algebra (matrices, eigenvalues, etc.) (05C50) Infinite graphs (05C63) Random walks on graphs (05C81)

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