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Quantum walks with an anisotropic coin I: spectral theory

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Abstract

We perform the spectral analysis of the evolution operator U of quantum walks with an anisotropic coin, which include one-defect models, two-phase quantum walks, and topological phase quantum walks as special cases. In particular, we determine the essential spectrum of U, we show the existence of locally U-smooth operators, we prove the discreteness of the eigenvalues of U outside the thresholds, and we prove the absence of singular continuous spectrum for U. Our analysis is based on new commutator methods for unitary operators in a two-Hilbert spaces setting, which are of independent interest.

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Acknowledgements

R. Tiedra de Aldecoa thanks the Graduate School of Mathematics of Nagoya University for its warm hospitality in January–February 2017. The authors also thank the anonymous referee for the valuable comments and for pointing out missing references which have been added.

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Authors and Affiliations

  1. Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya, 464-8602, Japan

    S. Richard

  2. Division of Mathematics and Physics, Faculty of Engineering, Shinshu University, Wakasato, Nagano, 380-8553, Japan

    A. Suzuki

  3. Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Av. Vicuña Mackenna 4860, Santiago, Chile

    R. Tiedra de Aldecoa

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  1. S. Richard
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Correspondence to R. Tiedra de Aldecoa.

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S. Richard was supported by JSPS Grant-in-Aid for Young Scientists A no 26707005, and on leave of absence from Univ. Lyon, Université Claude Bernard Lyon 1, CNRS UMR 5208, Institut Camille Jordan, 43 blvd. du 11 novembre 1918, F-69622 Villeurbanne cedex, France.

A. Suzuki was supported by JSPS Grant-in-Aid for Young Scientists B no 26800054.

R. Tiedra de Aldecoa was supported by the Chilean Fondecyt Grant 1170008.

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Richard, S., Suzuki, A. & Tiedra de Aldecoa, R. Quantum walks with an anisotropic coin I: spectral theory. Lett Math Phys 108, 331–357 (2018). https://doi.org/10.1007/s11005-017-1008-1

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