Spectral continuity for aperiodic quantum systems: applications of a folklore theorem. (English) Zbl 1472.82015
The authors provide a necessary and sufficient condition for a subshift to admit periodic approximations in the Hausdorff topology. Moreover, a rigorous justification for the accuracy and reliability of algorithmic methods that are used to numerically compute the spectra of certain self-adjoint operators, namely Hamiltonians associated with subshifts that admit periodic approximations, is given.
Reviewer: Maximilian Pechmann (Knoxville)
MSC:
| 82B44 | Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics |
| 82B20 | Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics |
| 35Q55 | NLS equations (nonlinear Schrödinger equations) |
Keywords:
aperiodic quantum systemsReferences:
| [1] | Allouche, J.-P.; Shallit, J. O., Complexité des suites de Rudin-Shapiro généralisées, Théorie Nombres, Bordeaux, 5, 283-302 (1993) · Zbl 0817.11014 · doi:10.5802/jtnb.94 |
| [2] | Allouche, J.-P., Schrödinger operators with Rudin-Shapiro potentials are not palindromic. Quantum problems in condensed matter physics, J. Math. Phys., 38, 1843-1848 (1997) · Zbl 0999.81018 · doi:10.1063/1.531916 |
| [3] | Anderson, J. E.; Putnam, I. F., Topological invariants for substitution tilings and their associated C^*-algebra, Ergodic Theory Dyn. Syst., 18, 509-537 (1998) · Zbl 1053.46520 · doi:10.1017/s0143385798100457 |
| [4] | Axel, F.; Gratias, D., Beyond Quasicrystals (1995), Springer-Verlag Berlin Heidelberg |
| [5] | Beckus, S., Spectral approximation of aperiodic Schrödinger operators (2016), Friedrich-Schiller-Universität Jena |
| [6] | Beckus, S.; Bellissard, J., Continuity of the spectrum of a field of self-adjoint operators, Ann. Henri Poincaré, 17, 3425-3442 (2016) · Zbl 1354.81018 · doi:10.1007/s00023-016-0496-3 |
| [7] | Beckus, S.; Bellissard, J.; de Nittis, G., Spectral continuity for aperiodic quantum systems I. General theory, J. Funct. Anal., 275, 2917-2977 (2018), 10.1016/j.jfa.2018.09.004; Beckus, S.; Bellissard, J.; de Nittis, G., Spectral continuity for aperiodic quantum systems I. General theory, J. Funct. Anal., 275, 2917-2977 (2018); Corrigendum 277(9), 3351-3353 (2019)., 10.1016/j.jfa.2018.09.004; · Zbl 1447.81107 |
| [8] | Beckus, S.; Bellissard, J.; Cornean, H., Hölder continuity of the spectra for aperiodic Hamiltonians, Ann. Henri Poincaré, 20, 3603-3631 (2019) · Zbl 1425.81038 · doi:10.1007/s00023-019-00848-6 |
| [9] | Beckus, S.; Pogorzelski, F., Delone dynamical systems and spectral convergence, Ergodic Theory Dyn. Syst., 40, 1510-1544 (2018) · Zbl 1443.37007 · doi:10.1017/etds.2018.116 |
| [10] | Bellissard, J.; Dorlas, T. C.; Hugenholtz, M. N.; Winnink, M., K-theory of C^*-algebras in solid state physics, Statistical Mechanics and Field Theory: Mathematical Aspects, 99-156 (1986), Springer-Verlag, Berlin · Zbl 0612.46066 |
| [11] | Bellissard, J.; Iochum, B.; Scoppola, E.; Testard, D., Spectral properties of one-dimensional quasi-crystals, Commun. Math. Phys., 125, 3, 527-543 (1989) · Zbl 0825.58010 · doi:10.1007/bf01218415 |
| [12] | Bellissard, J., Spectral properties of Schrödinger’s operator with a Thue-Morse potential, Number Theory and Physics, 140-150 (1990), Springer: Springer, Berlin · Zbl 0719.58038 |
| [13] | Bellissard, J.; Bovier, A.; Ghez, J.-M., Spectral properties of a tight binding Hamiltonian with period doubling potential, Commun. Math. Phys., 135, 379-399 (1991) · Zbl 0726.58038 · doi:10.1007/bf02098048 |
| [14] | Bellissard, J.; Iochum, B.; Testard, D., Continuity properties of the electronic spectrum of 1D quasicrystals, Commun. Math. Phys., 141, 353-380 (1991) · Zbl 0754.46049 · doi:10.1007/bf02101510 |
| [15] | Bellissard, J.; Garbaczewski, P.; Olkiewicz, R., Coherent and dissipative transport in aperiodic solids, Dynamics of Dissipation, 413-486 (2003), Springer |
| [16] | Bellissard, J., Bloch theory for 1-D FLC aperiodic media |
| [17] | Bellissard, J.; Kellendonk, J.; Lenz, D.; Savinien, J., Delone sets and material science: A program, Mathematics of Aperiodic Order (2015), Birkhäuser · Zbl 1373.82003 |
| [18] | Benza, V. G.; Sire, C., Electronic spectrum of the octagonal quasicrystal: Chaos, gaps and level clustering, Phys. Rev. B, 44, 10343-10345 (1991) · doi:10.1103/physrevb.44.10343 |
| [19] | Berkolaiko, G.; Kuchment, P., Introduction to Quantum Graphs (2013), American Mathematical Society · Zbl 1318.81005 |
| [20] | Berthé, V.; Rigo, M., Combinatorics, Automata and Number Theory (2010), Cambridge University Press: Cambridge University Press, Cambridge · Zbl 1197.68006 |
| [21] | Bovier, A.; Ghez, J.-M., Spectral properties of one-dimensional Schrödinger operators with potentials generated by substitutions, Commun. Math. Phys., 158, 45-66 (1993) · Zbl 0820.35099 · doi:10.1007/bf02097231 |
| [22] | de Bruijn, N. G., A combinatorial problem, Nederl. Akad. Wetensch, 49, 758-764 (1946) · Zbl 0060.02701 |
| [23] | Carmona, R.; Klein, A.; Martinelli, F., Anderson localization for Bernoulli and other singular potentials, Commun. Math. Phys., 108, 41-66 (1987) · Zbl 0615.60098 · doi:10.1007/bf01210702 |
| [24] | Casdagli, M., Symbolic dynamics for the renormalization map of a quasiperiodic Schrödinger equation, Commun. Math. Phys., 107, 295-318 (1986) · Zbl 0606.39004 · doi:10.1007/bf01209396 |
| [25] | Cassaigne, J.; Nicolas, F., Factor complexity, Combinatorics, Automata and Number Theory, 163-247 (2010), Cambridge University Press: Cambridge University Press, Cambridge · Zbl 1216.68204 |
| [26] | Chabauty, C., Limite d’ensembles et géométrie des nombres, Bull. Soc. Math. Fr., 78, 143-151 (1950) · Zbl 0039.04101 · doi:10.24033/bsmf.1412 |
| [27] | Coven, E. M.; Hedlund, G. A., Sequences with minimal block growth, Math. Syst. Theory, 7, 138-153 (1973) · Zbl 0256.54028 · doi:10.1007/bf01762232 |
| [28] | Castaing, C.; Valadier, M., Convex Analysis and Measurable Multifunctions (1977), Springer-Verlag: Springer-Verlag, Berlin · Zbl 0346.46038 |
| [29] | Damanik, D., Singular continuous spectrum for the period doubling Hamiltonian on a set of full measure, Commun. Math. Phys., 196, 2, 477-483 (1998) · Zbl 0907.34074 · doi:10.1007/s002200050432 |
| [30] | Damanik, D., Local symmetries in the period-doubling sequence, Discrete Appl. Math., 100, 115-121 (2000) · Zbl 0943.68127 · doi:10.1016/s0166-218x(99)00199-7 |
| [31] | Damanik, D., Substitution Hamiltonians with bounded trace map orbits, J. Math. Anal. Appl., 249, 393-411 (2000) · Zbl 0996.37011 · doi:10.1006/jmaa.2000.6876 |
| [32] | Damanik, D.; Lenz, D., Uniform spectral properties of one-dimensional quasicrystals. IV. Quasi-Sturmian potentials, J. Anal. Math., 90, 115-139 (2003) · Zbl 1173.81315 · doi:10.1007/bf02786553 |
| [33] | Damanik, D.; Lenz, D., Zero-measure Cantor spectrum for Schrödinger operators with low-complexity potentials, J. Math. Pures Appl., 85, 671-686 (2006) · Zbl 1122.47028 · doi:10.1016/j.matpur.2005.11.002 |
| [34] | Damanik, D., Schrödinger operators with dynamically defined potentials: A survey, Ergodic Theory Dynam. Systems, 37, 6, 1681-1764 (2017) · Zbl 1541.81060 · doi:10.1017/etds.2015.120 |
| [35] | Damanik, D.; Embree, M.; Gorodetski, A.; Tcheremchantsev, S., The fractal dimension of the spectrum of the Fibonacci Hamiltonian, Commun. Math. Phys., 280, 499-516 (2008) · Zbl 1192.81151 · doi:10.1007/s00220-008-0451-3 |
| [36] | Damanik, D.; Embree, M.; Gorodetski, A.; Kellendonk, J.; Lenz, D.; Savinien, J., Spectral properties of Schrödinger operators arising in the study of quasicrystals, Mathematics of Aperiodic Order (2015), Birkhäuser · Zbl 1378.81031 |
| [37] | Damanik, D.; Gorodetski, A.; Solomyak, B., Absolutely continuous convolutions of singular measures and an application to the square Fibonacci Hamiltonian, Duke Math. J., 164, 1603-1640 (2015) · Zbl 1358.37117 · doi:10.1215/00127094-3119739 |
| [38] | Damanik, D.; Gorodetski, A.; Yessen, W., The Fibonacci Hamiltonian, Invent. Math., 206, 629-692 (2016) · Zbl 1359.81108 · doi:10.1007/s00222-016-0660-x |
| [39] | Duneau, M.; Mosseri, R.; Oguey, C., Approximants of quasiperiodic structures generated by the inflation mapping, J. Phys. A: Math. Gen., 22, 4549-4564 (1989) · Zbl 0719.52015 · doi:10.1088/0305-4470/22/21/017 |
| [40] | Fell, J. M. G., A Hausdorff topology for the closed subsets of a locally compact non-Hausdorff space, Proc. Am. Math. Soc., 13, 472-476 (1962) · Zbl 0106.15801 · doi:10.1090/s0002-9939-1962-0139135-6 |
| [41] | Fiorenzi, F., Periodic configurations of subshifts on groups, Int. J. Algebra Comput., 19, 315-335 (2009) · Zbl 1205.37026 · doi:10.1142/s0218196709005123 |
| [42] | Frisch, J.; Tamuz, O., Symbolic dynamics on amenable groups: The entropy of generic shifts, Ergodic Theory Dyn. Syst., 37, 1187-1210 (2017) · Zbl 1386.37013 · doi:10.1017/etds.2015.84 |
| [43] | Fogg, P., Substitutions in Dynamics, Arithmetics and Combinatorics (2002), Springer · Zbl 1014.11015 |
| [44] | Gähler, F. |
| [45] | Golay, M. J. E., Static multilist spectrometry and its application to panoramic display of infrared spectra, J. Opt. Soc. Am., 41, 468-472 (1951) · doi:10.1364/josa.41.000468 |
| [46] | Greenleaf, F. P., Invariant Means on Topological Groups and Their Applications (1969), Van Nostrand Reinhold · Zbl 0174.19001 |
| [47] | Hedlund, G. A.; Morse, M., Symbolic dynamics II: Sturmian trajectories, Am. J. Math., 62, 1-42 (1940) · Zbl 0022.34003 · doi:10.2307/2371431 |
| [48] | Hochman, M., Genericity in topological dynamics, Ergodic Theory Dyn. Syst., 28, 125-165 (2008) · Zbl 1171.37305 · doi:10.1017/s0143385707000521 |
| [49] | Hof, A.; Knill, O.; Simon, B., Singular continuous spectrum for palindromic Schrödinger operators, Commun. Math. Phys., 174, 149-159 (1995) · Zbl 0839.11009 · doi:10.1007/bf02099468 |
| [50] | One of the authors, J.B. thanks André Katz for this conversation in the early 1990’s where the defects that can occur in the limit correspond to what Physicists call worms in a quasicrystal. |
| [51] | Kellendonk, J.; Prodan, E., Bulk-boundary correspondance for Sturmian Kohmoto like models, Ann. Henri Poincaré, 20, 2039-2070 (2019) · Zbl 1492.82024 · doi:10.1007/s00023-019-00792-5 |
| [52] | Kohmoto, M.; Kadanoff, L. P.; Tang, C., Localization problem in one dimension: Mapping and escape, Phys. Rev. Lett., 50, 1870-1872 (1983) · doi:10.1103/physrevlett.50.1870 |
| [53] | Kohmoto, M.; Oono, Y., Cantor spectrum for an almost periodic Schrödinger equation and a dynamical map, Phys. Lett. A, 102, 145-148 (1984) · doi:10.1016/0375-9601(84)90928-9 |
| [54] | Kotani, S., Jacobi matrices with random potentials taking finitely many values, Rev. Math. Phys., 01, 129-133 (1989) · Zbl 0713.60074 · doi:10.1142/s0129055x89000067 |
| [55] | Lenz, D., Singular spectrum of Lebesgue measure zero for one-dimensional quasicrystal, Commun. Math. Phys., 227, 119-130 (2002) · Zbl 1065.47035 · doi:10.1007/s002200200624 |
| [56] | Lind, D.; Marcus, B., An Introduction to Symbolic Dynamics and Coding (1995), Cambridge University Press: Cambridge University Press, Cambridge · Zbl 1106.37301 |
| [57] | Liu, Q.-H.; Qu, Y.-H., Uniform convergence of Schrödinger cocycles over simple Toeplitz subshift, Ann. Henri Poincaré, 12, 153-172 (2011) · Zbl 1211.81059 · doi:10.1007/s00023-010-0075-y |
| [58] | Liu, Q.; Qu, Y., On the Hausdorff dimension of the spectrum of the Thue-Morse Hamiltonian, Commun. Math. Phys., 338, 867-891 (2015) · Zbl 1333.47026 · doi:10.1007/s00220-015-2377-x |
| [59] | Liu, Q. H.; Qu, Y. H.; Yiao, X., Unbounded trace orbits of Thue-Morse Hamiltonian, J. Stat. Phys., 166, 6, 1509-1557 (2017) · Zbl 1376.82038 |
| [60] | Moreno, E., de Bruijn sequences and de Bruijn graphs for a general language, Inform. Process. Lett., 96, 6, 214-219 (2005) · Zbl 1184.68323 · doi:10.1016/j.ipl.2005.05.028 |
| [61] | Ostlund, S.; Pandit, R.; Rand, D.; Schellnhuber, H. J.; Siggia, E. D., One-dimensional Schrödinger equation with an almost periodic potential, Phys. Rev. Lett., 50, 1873-1876 (1983) · doi:10.1103/physrevlett.50.1873 |
| [62] | Ostlund, S.; Kim, S.-H., Renormalization of quasiperiodic mappings, Phys. Scr., T9, 193-198 (1985) · Zbl 1063.37526 · doi:10.1088/0031-8949/1985/t9/031 |
| [63] | Petite, S., Mike Hochman, (Hebrew University, Jerusalem) as one of these experts |
| [64] | Queffélec, M., Substitution Dynamical Systems: Spectral Analysis (1987), Springer-Verlag: Springer-Verlag, Berlin · Zbl 0642.28013 |
| [65] | Queffélec, M., Substitution Dynamical Systems: Spectral Analysis (2010), Springer-Verlag: Springer-Verlag, Berlin · Zbl 1225.11001 |
| [66] | Rauzy, G., Suites à termes dans un alphabet fini (1983), University of Bordeaux I: University of Bordeaux I, Talence · Zbl 0547.10048 |
| [67] | Rudin, W., Some theorems on Fourier coefficients, Proc. Am. Math. Soc., 10, 855-859 (1959) · Zbl 0091.05706 · doi:10.1090/s0002-9939-1959-0116184-5 |
| [68] | Sadun, L., Tiling spaces are inverse limits, J. Math. Phys., 44, 11, 5410-5414 (2003) · Zbl 1063.37019 · doi:10.1063/1.1613041 |
| [69] | Shapiro, H. S., Extremal problems for polynomials and power series (1951), Massachusetts Institute of Technology: Massachusetts Institute of Technology, Cambridge, MA |
| [70] | Shechtman, D.; Blech, I.; Gratias, D.; Cahn, J. W., Metallic phase with long-range orientational order and No translational symmetry, Phys. Rev. Lett., 53, 1951-1953 (1984) · doi:10.1103/physrevlett.53.1951 |
| [71] | Sütő, A., The spectrum of a quasi-periodic Schrödinger operator, Commun. Math. Phys., 111, 409-415 (1987) · Zbl 0624.34017 · doi:10.1007/bf01238906 |
| [72] | Sütő, A., Singular continuous spectrum on a Cantor set of zero Lebesgue measure for the Fibonacci Hamiltonian, J. Stat. Phys., 56, 525-531 (1989) · Zbl 0712.58046 · doi:10.1007/BF01044450 |
| [73] | Sütő, A., Schrödinger difference equation with deterministic ergodic potentials, Beyond Quasicrystals (1995), Springer-Verlag Berlin Heidelberg · Zbl 0884.34083 |
| [74] | Vietoris, L., Bereiche zweiter ordnung, Monatsh. Math. Phys., 32, 1, 258-280 (1922) · JFM 48.0205.02 |
| [75] | von Querenburg, B., Mengentheoretische Topologie (2001), Springer-Verlag: Springer-Verlag, Berlin, New York · Zbl 0960.54001 |
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.
