Abstract
We study Jacobi matrices that are uniformly approximated by periodic operators. We show that if the rate of approximation is sufficiently rapid, then the associated quantum dynamics are ballistic in a rather strong sense; namely, the (normalized) Heisenberg evolution of the position operator converges strongly to a self-adjoint operator that is injective on the space of absolutely summable sequences. In particular, this means that all transport exponents corresponding to well-localized initial states are equal to one. Our result may be applied to a class of quantum many-body problems. Specifically, we establish a lower bound on the Lieb–Robinson velocity for an isotropic XY spin chain on the integers with limit-periodic couplings.
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Ahlbrecht, A., Vogts, H., Werner, A., Werner, R.: Asymptotic evolution of quantum walks with random coin. J. Math. Phys. 52, 042201 (2011)
Asch J., Knauf A.: Motion in periodic potentials. Nonlinearity 11, 175–200 (1998)
Avila A.: On the spectrum and Lyapunov exponent of limit-periodic Schrödinger operators. Commun. Math. Phys. 288, 907–918 (2009)
Avron J., Simon B.: Almost periodic Schrödinger operators. I. Limit periodic potentials. Commun. Math. Phys. 82, 101–120 (1981)
Bellissard J., Schulz-Baldes H.: Subdiffusive quantum transport for 3D Hamiltonians with absolutely continuous spectra. J. Stat. Phys. 99, 587–594 (2000)
Bruneau L., Jakšić V., Pillet C.-A.: Landauer–Bütttiker formula and Schrödinger conjecture. Commun. Math. Phys. 319, 501–513 (2013)
Bruneau L., Jakšić V., Last Y., Pillet C.-A.: Landauer–Büttiker and Thouless conductance. Commun. Math. Phys. 338, 347–366 (2015)
Bruneau L., Jakšić V., Last Y., Pillet C.-A.: Conductance and absolutely continuous spectrum of 1D samples. Commun. Math. Phys. 344, 959–981 (2016)
Bruneau L., Jakšić V., Last Y., Pillet C.-A.: Crystalline conductance and absolutely continuous spectrum of 1D samples. Lett. Math. Phys. 106, 787–797 (2016)
Bruneau, L., Jakšić, V., Last, Y., Pillet, C.-A.: What is absolutely continuous spectrum? (Preprint). arXiv:1602.01893
Cantero M.-J., Grünbaum A., Moral L., Velázquez L.: Matrix-valued Szegő polynomials and quantum random walks. Commun. Pure Appl. Math. 63, 464–507 (2010)
Cantero M.-J., Grünbaum A., Moral L., Velázquez L.: The CGMV method for quantum walks. Quantum Inf. Process. 11, 1149–1192 (2012)
Carleson, L.: On \({H^\infty}\) in multiply connected domains. In: Harmonic Analysis. Conference in Honor of Antony Zygmund, vol. II, pp. 349–382 (1983)
Chulaevskii V.: Perturbations of a Schrödinger operator with periodic potential (Russian). Uspekhi Mat. Nauk 36, 203–204 (1981)
Chulaevsky V.: Almost Periodic Operators and Related Nonlinear Integrable Systems. Manchester University Press, Manchester (1989)
Damanik D., Fillman J., Ong D.C.: Spreading estimates for quantum walks on the integer lattice via power-law bounds on transfer matrices. J. Math. Pures Appl. 105, 293–341 (2016)
Damanik D., Gan Z.: Limit-periodic Schrödinger operators in the regime of positive Lyapunov exponents. J. Funct. Anal. 258, 4010–4025 (2010)
Damanik D., Gan Z.: Limit-periodic Schrödinger operators with uniformly localized eigenfunctions. J. Anal. Math. 115, 33–49 (2011)
Damanik D., Gan Z.: Spectral properties of limit-periodic Schrödinger operators. Commun. Pure Appl. Anal. (3) 10, 859–871 (2011)
Damanik D., Gorodetski A.: An extension of the Kunz-Souillard approach to localization in one dimension and applications to almost-periodic Schrödinger operators. Adv. Math. 297, 149–173 (2016)
Damanik D., Lukic M., Yessen W.: Quantum dynamics of periodic and limit-periodic Jacobi and block Jacobi matrices with applications to some quantum many body problems. Commun. Math. Phys. 337, 1535–1561 (2015)
Egorova, I.E.: Spectral analysis of Jacobi limit-periodic matrices. Dokl. Akad. Nauk Ukrain. SSR Ser. A 3, 7–9. (1987). (in Russian)
Fillman, J.: Spectral homogeneity of discrete one-dimensional limit-periodic operators. J. Spectral Theory arXiv:1409.7734. (to appear)
Fillman, J., Lukic, M.: Spectral homogeneity of limit-periodic Schrödinger operators. J. Spectral Theory (to appear)
Gan Z.: An exposition of the connection between limit-periodic potentials and profinite groups. Math. Model. Nat. Phenom. 5:4, 158–174 (2010)
Gan Z., Krüger H.: Optimality of log-Hölder continuity of the integrated density of states. Math. Nachr. 284, 1919–1923 (2011)
Kachkovskiy I.: On transport properties of isotropic quasiperiodic XY spin chains. Commun. Math. Phys. 345(2), 659–673 (2016)
Karpeshina, Y., Lee, Y.-R., Shterenberg, R., Stolz, G.: Ballistic transport for the Schrödinger operator with limit-periodic or quasi-periodic potential in dimension two (Preprint). arXiv:1507.06523
Last Y.: On the measure of gaps and spectra for discrete 1D Schrödinger operators. Commun. Math. Phys. 149, 347–360 (1992)
Last Y.: A relation between a.c. spectrum of ergodic Jacobi matrices and the spectra of periodic approximants. Commun. Math. Phys. 151, 183–192 (1993)
Last Y.: Quantum dynamics and decompositions of singular continuous spectra. J. Funct. Anal. 142, 406–445 (1996)
Last Y., Simon B.: Eigenfunctions, transfer matrices, and absolutely continuous spectrum of one-dimensional Schrödinger operators. Invent. Math. 135, 329–367 (1999)
Lieb E.H., Robinson D.W.: The finite group velocity of quantum spin systems. Commun. Math. Phys. 28, 251–257 (1972)
Lieb E.H., Schultz T., Mattis D.: Two soluble models of an antiferromagnetic chain. Ann. Phys. 16, 407–466 (1961)
Molchanov S.A., Chulaevskii V.: The structure of a spectrum of the lacunary-limit-periodic Schrödinger operator (Russian). Funktsional. Anal. i Prilozhen. 18, 90–91 (1984)
Moser J.: An example of a Schrödinger equation with almost periodic potential and nowhere dense spectrum. Comment. Math. Helv. 56, 198–224 (1981)
Nachtergaele, B., Sims, R.: Locality estimates for quantum spin systems. Siboravičius, V. (ed) New Trends in Mathematical Physics, pp. 591–614. Springer, Berlin (2009)
Ong D.: Limit-periodic Verblunsky coefficients for orthogonal polynomials on the unit circle. J. Math. Anal. Appl. 394(2), 633–644 (2012)
Pastur L., Tkachenko V.A.: On the spectral theory of the one-dimensional Schrödinger operator with limit-periodic potential (Russian). Dokl. Akad. Nauk SSSR. 279, 1050–1053 (1984)
Pastur L., Tkachenko V.A.: Spectral theory of a class of one-dimensional Schrödinger operators with limit-periodic potentials (Russian). Trudy Moskov. Mat. Obshch. 51, 114–168 (1988)
Poltoratski A., Remling C.: Reflectionless Herglotz functions and Jacobi matrices. Commun. Math. Phys. 288, 1007–1021 (2009)
Pöschel J.: Examples of discrete Schrödinger operators with pure point spectrum. Commun. Math. Phys. 88, 447–463 (1983)
Remling C.: The absolutely continuous spectrum of Jacobi matrices. Ann. Math. 174, 125–171 (2011)
Simon B.: Kotani theory for one-dimensional stochastic Jacobi matrices. Commun. Math. Phys. 89, 227–234 (1983)
Simon, B.: Szegö’s Theorem and its descendants: spectral theory for l 2 perturbations of orthogonal polynomials. In: M.B. Porter Lectures. Princeton University Press, Princeton (2011)
Teschl, G.: Jacobi operators and completely integrable nonlinear lattices. In: Mathematical Surveys and Monographs, vol. 72. American Mathematical Society, Providence (2000)
Zhang, Z., Zhao, Z.: Ballistic transport and absolute continuity of one-frequency Schrödinger operators (Preprint). arXiv:1512.02195
Zhao Z.: Ballistic motion in one-dimensional quasi-periodic discrete Schrödinger equation. Commun. Math. Phys. 347(2), 511–549 (2016)
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Fillman, J. Ballistic Transport for Limit-Periodic Jacobi Matrices with Applications to Quantum Many-Body Problems. Commun. Math. Phys. 350, 1275–1297 (2017). https://doi.org/10.1007/s00220-016-2785-6
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DOI: https://doi.org/10.1007/s00220-016-2785-6