Eigenvalues, absolute continuity and localizations for periodic unitary transition operators

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3 Citations (Scopus)

Abstract

The localization phenomenon for periodic unitary transition operators on a Hilbert space consisting of square summable functions on an integer lattice with values in a finite-dimensional Hilbert space, which is a generalization of the discrete-time quantum walks with constant coin matrices, is discussed. It is proved that a periodic unitary transition operator has an eigenvalue if and only if the corresponding unitary matrix-valued function on a torus has an eigenvalue which does not depend on the points on the torus. It is also proved that the continuous spectrum of a periodic unitary transition operator is absolutely continuous. As a result, it is shown that the localization happens if and only if there exists an eigenvalue, and when there exists only one eigenvalue, the long-time limit of transition probabilities coincides with the point-wise norm of the projection of the initial state to the eigenspace. The results can be applied to certain unitary operators on a Hilbert space on a covering graph, called a topological crystal, over a finite graph. An analytic perturbation theory for matrices in several complex variables is employed to show the result about absolute continuity for periodic unitary transition operators.

Original languageEnglish
Article number1950011
JournalInfinite Dimensional Analysis, Quantum Probability and Related Topics
Volume22
Issue number2
DOIs
Publication statusPublished - 2019 Jun 1

Keywords

  • Quantum walks
  • absolutely continuous spectrum
  • localizations
  • periodic unitary transition operators

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Statistics and Probability
  • Mathematical Physics
  • Applied Mathematics

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