Eigenvalues, absolute continuity and localizations for periodic unitary transition operators

研究成果: Article査読

3 被引用数 (Scopus)

抄録

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.

本文言語English
論文番号1950011
ジャーナルInfinite Dimensional Analysis, Quantum Probability and Related Topics
22
2
DOI
出版ステータスPublished - 2019 6 1

ASJC Scopus subject areas

  • 統計物理学および非線形物理学
  • 統計学および確率
  • 数理物理学
  • 応用数学

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