TY - JOUR

T1 - Eigenvalues, absolute continuity and localizations for periodic unitary transition operators

AU - Tate, Tatsuya

N1 - Funding Information:
The author is partially supported by JSPS KAKENHI Grant Nos. 25400068, 18K03267, 24340031, 15H02055 and 17H06465.
Publisher Copyright:
© 2019 The Author(s).

PY - 2019/6/1

Y1 - 2019/6/1

N2 - 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.

AB - 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.

KW - Quantum walks

KW - absolutely continuous spectrum

KW - localizations

KW - periodic unitary transition operators

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U2 - 10.1142/S0219025719500115

DO - 10.1142/S0219025719500115

M3 - Article

AN - SCOPUS:85069712613

VL - 22

JO - Infinite Dimensional Analysis, Quantum Probability and Related Topics

JF - Infinite Dimensional Analysis, Quantum Probability and Related Topics

SN - 0219-0257

IS - 2

M1 - 1950011

ER -