Dispersive estimates for quantum walks on 1D lattice. (English) Zbl 1485.35329

The paper focuses on the study in the long time of quantum walks (QWs) within coins that converge to some constant coin \(C_0\) as \(|x|\rightarrow \infty\). The core tool used to tackle the \(\ell^\infty-\ell^1-\) decay estimate provided by eq. (1.1) – see Theorem 1.9. – are centered on the notion of Jost solutions (Section 5).
To this end, the authors take into account the so-called Cantero-Moral-Velázquez (CMV) representation of QWs (see [M. J. Cantero et al., Commun. Pure Appl. Math. 63, No. 4, 464–507 (2010; Zbl 1186.81036)]) to describe the pair \((\lambda,u)\) of solutions of \((U-e^{i\lambda})u=0\) through the the transfer matrix representation of the eigenvalue problem \((U-\lambda)u=0\) (see Section 3).
The paper under consideration is situated on the crossroads of orthogonal polynomials and spectral analysis. And it involves also the framework of Pauli matrices, an essential tool to describe the symmetries of the transfer matrix of the aforementioned eigenvalue problem. For the interested reader, this paper shall be considered as a starting paper on the scope of QWs. Indeed, the proofs enclosed are structured, self-contained and easy to reach.


35Q41 Time-dependent Schrödinger equations and Dirac equations
81U30 Dispersion theory, dispersion relations arising in quantum theory
82B41 Random walks, random surfaces, lattice animals, etc. in equilibrium statistical mechanics


Zbl 1186.81036
Full Text: DOI arXiv


[1] A. Ambainis, E. Bach, A. Nayak, A. Vishwanath and J. Watrous, One-dimensional quantum walks, In: Proceedings of the Thirty-Third Annual ACM Symposium on Theory of Computing, ACM, New York, 2001, 37-49. · Zbl 1323.81021
[2] A. Ambainis, J. Kempe and A. Rivosh, Coins make quantum walks faster, In: Proceedings of the Sixteenth Annual ACM-SIAM Symposium on Discrete Algorithms, ACM, New York, 2005, 1099-1108. · Zbl 1297.68076
[3] J. K. Asbóth and H. Obuse, Bulk-boundary correspondence for chiral symmetric quantum walks, Phys. Rev. B, 88 (2013), 121406.
[4] J. Asch, O. Bourget and A. Joye, Spectral stability of unitary network models, Rev. Math. Phys., 27 (2015), no. 7, 1530004, 22 pp. · Zbl 1326.81035
[5] D. Bambusi, Asymptotic stability of breathers in some Hamiltonian networks of weakly coupled oscillators, Comm. Math. Phys., 324 (2013), 515-547. · Zbl 1286.37064
[6] M. J. Cantero, F. A. Grünbaum, L. Moral and L. Velázquez, Matrix-valued Szegő polynomials and quantum random walks, Comm. Pure Appl. Math., 63 (2010), 464-507. · Zbl 1186.81036
[7] M. J. Cantero, L. Moral and L. Velázquez, Five-diagonal matrices and zeros of orthogonal polynomials on the unit circle, Linear Algebra Appl., 362 (2003), 29-56. · Zbl 1022.42013
[8] C. Cedzich, F. A. Grünbaum, C. Stahl, L. Velázquez, A. H. Werner and R. F. Werner, Bulk-edge correspondence of one-dimensional quantum walks, J. Phys. A, 49 (2016), no. 21, 21LT01. · Zbl 1342.82064
[9] A. M. Childs, Universal computation by quantum walk, Phys. Rev. Lett., 102 (2009), no. 18, 180501, 4 pp.
[10] S. Cuccagna and M. Tarulli, On asymptotic stability of standing waves of discrete Schrödinger equation in \(\mathbb{Z} \), SIAM J. Math. Anal., 41 (2009), 861-885. · Zbl 1189.35303
[11] I. Egorova, E. Kopylova and G. Teschl, Dispersion estimates for one-dimensional discrete Schrödinger and wave equations, J. Spectr. Theory, 5 (2015), 663-696. · Zbl 1332.35313
[12] S. Endo, T. Endo, N. Konno, E. Segawa and M. Takei, Weak limit theorem of a two-phase quantum walk with one defect, Interdiscip. Inform. Sci., 22 (2016), 17-29. · Zbl 1453.62406
[13] T. Endo, N. Konno, H. Obuse and E. Segawa, Sensitivity of quantum walks to a boundary of two-dimensional lattices: approaches based on the CGMV method and topological phases, J. Phys. A, 50 (2017), no. 45, 455302. · Zbl 1386.82022
[14] R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integrals, emended edition, Dover Publications, Inc., Mineola, NY, 2010, emended and with a preface by D. F. Styer. · Zbl 1220.81156
[15] J. Fillman and D. C. Ong, Purely singular continuous spectrum for limit-periodic CMV operators with applications to quantum walks, J. Funct. Anal., 272 (2017), 5107-5143. · Zbl 1397.81115
[16] T. Fuda, D. Funakawa and A. Suzuki, Localization for a one-dimensional split-step quantum walk with bound states robust against perturbations, J. Math. Phys., 59 (2018), 082201. · Zbl 1395.81143
[17] T. Fuda, D. Funakawa and A. Suzuki, Weak limit theorem for a one-dimensional split-step quantum walk, Rev. Roumaine Math. Pures Appl., 64 (2019), 157-165. · Zbl 1438.47023
[18] G. Grimmett, S. Janson and P. F. Scudo, Weak limits for quantum random walks, Phys. Rev. E, 69 (2004), 026119.
[19] Y. Gerasimenko, B. Tarasinski and C. W. J. Beenakker, Attractor-repeller pair of topological zero modes in a nonlinear quantum walk, Phys. Rev. A, 93 (2016), 022329.
[20] F. Gesztesy and M. Zinchenko, Weyl-Titchmarsh theory for CMV operators associated with orthogonal polynomials on the unit circle, J. Approx. Theory, 139 (2006), 172-213. · Zbl 1118.47023
[21] I. Gohberg, S. Goldberg and M. A. Kaashoek, Classes of Linear Operators Vol. I, Oper. Theory Adv. Appl., 49, Birkhäuser Verlag, Basel, 1990. · Zbl 0745.47002
[22] M. Goldberg and W. Schlag, Dispersive estimates for Schrödinger operators in dimensions one and three, Comm. Math. Phys., 251 (2004), 157-178. · Zbl 1086.81077
[23] D. Gross, V. Nesme, H. Vogts and R. F. Werner, Index theory of one dimensional quantum walks and cellular automata, Comm. Math. Phys., 310 (2012), 419-454. · Zbl 1238.81154
[24] S. P. Gudder, Quantum Probability, Probab. Math. Statist., Academic Press, Inc., Boston, MA, 1988. · Zbl 0653.60004
[25] J.-L. Journé, A. Soffer and C. D. Sogge, Decay estimates for Schrödinger operators, Comm. Pure Appl. Math., 44 (1991), 573-604. · Zbl 0743.35008
[26] M. Karski, L. Förster, J.-M. Choi, A. Steffen, W. Alt, D. Meschede and A. Widera, Quantum walk in position space with single optically trapped atoms, Science, 325 (2009), no. 5937, 174-177.
[27] Y. Katznelson, An Introduction to Harmonic Analysis, third edition, Cambridge Math. Lib., Cambridge Univ. Press, Cambridge, 2004. · Zbl 1055.43001
[28] M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980. · Zbl 0922.35028
[29] P. G. Kevrekidis, D. E. Pelinovsky and A. Stefanov, Asymptotic stability of small bound states in the discrete nonlinear Schrödinger equation, SIAM J. Math. Anal., 41 (2009), 2010-2030. · Zbl 1197.35270
[30] A. Kitaev, Anyons in an exactly solved model and beyond, Ann. Physics, 321 (2006), 2-111. · Zbl 1125.82009
[31] T. Kitagawa, Topological phenomena in quantum walks: elementary introduction to the physics of topological phases, Quantum Inf. Process., 11 (2012), 1107-1148. · Zbl 1252.82088
[32] T. Kitagawa, M. S. Rudner, E. Berg and E. Demler, Exploring topological phases with quantum walks, Phys. Rev. A, 82 (2010), 033429.
[33] N. Konno, Quantum random walks in one dimension, Quantum Inf. Process., 1 (2002), 345-354. · Zbl 1329.82012
[34] C.-W. Lee, P. Kurzyński and H. Nha, Quantum walk as a simulator of nonlinear dynamics: Nonlinear dirac equation and solitons, Phys. Rev. A, 92 (2015), 052336.
[35] M. Maeda, H. Sasaki, E. Segawa, A. Suzuki and K. Suzuki, Scattering and inverse scattering for nonlinear quantum walks, Discrete Contin. Dyn. Syst., 38 (2018), 3687-3703. · Zbl 1395.81111
[36] M. Maeda, H. Sasaki, E. Segawa, A. Suzuki and K. Suzuki, Weak limit theorem for a nonlinear quantum walk, Quantum Inf. Process., 17 (2018), art. no. 215. · Zbl 1398.81141
[37] M. Maeda, H. Sasaki, E. Segawa, A. Suzuki and K. Suzuki, Dynamics of solitons for nonlinear quantum walks, J. Physics Communications, 3 (2019), no. 7, 075002. · Zbl 1395.81111
[38] K. Manouchehri and J. Wang, Physical Implementation of Quantum Walks, Quantum Sci. Technol., Springer, Heidelberg, 2014. · Zbl 1278.81010
[39] D. A. Meyer, From quantum cellular automata to quantum lattice gases, J. Statist. Phys., 85 (1996), 551-574. · Zbl 0952.37501
[40] T. Mizumachi and D. Pelinovsky, On the asymptotic stability of localized modes in the discrete nonlinear Schrödinger equation, Discrete Contin. Dyn. Syst. Ser. S, 5 (2012), 971-987. · Zbl 1260.37047
[41] C. Navarrete-Benlloch, A. Pérez and E. Roldán, Nonlinear optical galton board, Phys. Rev. A, 75 (2007), 062333.
[42] D. E. Pelinovsky and A. Stefanov, On the spectral theory and dispersive estimates for a discrete Schrödinger equation in one dimension, J. Math. Phys., 49 (2008), no. 11, 113501, 17 pp. · Zbl 1159.81336
[43] R. Portugal, Quantum Walks and Search Algorithms, Quantum Sci. Technol., Springer, New York, 2013. · Zbl 1275.81004
[44] A. Schreiber, K. N. Cassemiro, V. Potoček, A. Gábris, I. Jex and C. Silberhorn, Photonic quantum walks in a fiber based recursion loop, In: Quantum Communication, Measurement and Computing (QCMC), AIP Conference Proceedings, 1363, AIP Publ., 2011, 155-158.
[45] A. Schreiber, A. Gábris, P. P. Rohde, K. Laiho, M. Štefaňák, V. Potoček, C. Hamilton, I. Jex and C. Silberhorn, A 2d quantum walk simulation of two-particle dynamics, Science, 336 (2012), no. 6077, 55-58.
[46] E. Segawa and A. Suzuki, Generator of an abstract quantum walk, Quantum Stud. Math. Found., 3 (2016), 11-30. · Zbl 1338.81260
[47] S. Richard, A. Suzuki and R. Tiedra de Aldecoa, Quantum walks with an anisotropic coin I: spectral theory, Lett. Math. Phys., 108 (2018), 331-357. · Zbl 1384.81033
[48] S. Richard, A. Suzuki and R. Tiedra de Aldecoa, Quantum walks with an anisotropic coin II: scattering theory, Lett. Math. Phys., 109 (2019), 61-88. · Zbl 1411.81215
[49] A. Soffer and M. I. Weinstein, Multichannel nonlinear scattering for nonintegrable equations II. The case of anisotropic potentials and data, J. Differential Equations, 98 (1992), 376-390. · Zbl 0795.35073
[50] A. Stefanov and P. G. Kevrekidis, Asymptotic behaviour of small solutions for the discrete nonlinear Schrödinger and Klein-Gordon equations, Nonlinearity, 18 (2005), 1841-1857. · Zbl 1181.35266
[51] E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Math. Ser., 43, Princeton Univ. Press, Princeton, NJ, 1993, with the assistance of T. S. Murphy, Monographs in Harmonic Analysis, III. · Zbl 0821.42001
[52] E. M. Stein and R. Shakarchi, Complex Analysis, Princeton Lect. Anal., 2, Princeton Univ. Press, Princeton, NJ, 2003. · Zbl 1020.30001
[53] T. Sunada and T. Tate, Asymptotic behavior of quantum walks on the line, J. Funct. Anal., 262 (2012), 2608-2645. · Zbl 1241.82042
[54] A. Suzuki, Asymptotic velocity of a position-dependent quantum walk, Quantum Inf. Process., 15 (2016), 103-119. · Zbl 1333.81214
[55] R. Weder, \(L^p-L^{\dot{p}}\) estimates for the Schrödinger equation on the line and inverse scattering for the nonlinear Schrödinger equation with a potential, J. Funct. Anal., 170 (2000), 37-68. · Zbl 0943.34070
[56] F. Zähringer, G. Kirchmair, R. Gerritsma, E. Solano, R. Blatt and C. F. Roos, Realization of a quantum walk with one and two trapped ions, Phys. Rev. Lett., 104 (2010), 100503.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.