Tridiagonal Substitution Hamiltonians

M. Mei¹ ; W. Yessen²

¹ Mathematics & Computer Science, Denison University, Granville, OH 43023-0810
² Mathematics, Rice University, 1600 Main St. MS-136, Houston, TX 77005

Mathematical modelling of natural phenomena, Tome 9 (2014) no. 5, pp. 204-238.

Résumé

We consider a family of discrete Jacobi operators on the one-dimensional integer lattice with Laplacian and potential terms modulated by a primitive invertible two-letter substitution. We investigate the spectrum and the spectral type, the fractal structure and fractal dimensions of the spectrum, exact dimensionality of the integrated density of states, and the gap structure. We present a review of previous results, some applications, and open problems. Our investigation is based largely on the dynamics of trace maps. This work is an extension of similar results on Schrödinger operators, although some of the results that we obtain differ qualitatively and quantitatively from those for the Schrödinger operators. The nontrivialities of this extension lie in the dynamics of the associated trace map as one attempts to extend the trace map formalism from the Schrödinger cocycle to the Jacobi one. In fact, the Jacobi operators considered here are, in a sense, a test item, as many other models can be attacked via the same techniques, and we present an extensive discussion on this.

Détail

DOI : 10.1051/mmnp/20149514

Bibliographie

[1] S. Astels Trans. Amer. Math. Soc. 2000 133 170

[2] A. Avila, S. Jitomirskaya Annal. Math. 2009 303 342

[3] J. Avron, V. Mouche, P. H. M., B. Simon Commun. Math. Phys. 1990 103 118

[4] S. Beckus, F. Pogorzelski Mathematical Physics, Analysis and Geometry 2013 289 308

[5] J. Bellissard. Spectral properties of Schrödinger’s operator with a Thue-Morse potential. Number Theory and Physics (Les Houches, 1989), 140–150, Springer Proc. Phys., 47, Springer, Berlin 1990.

[6] J. Bellissard, A. Bovier, J.-M. Ghez Commun. Math. Phys. 1991 379 399

[7] J. Bellissard, A. Bovier, J.-M. Ghez Rev. Math. Phys. 1992 1 37

[8] J. Bellissard, B. Iochum, E. Scoppola, D. Testard Commun. Math. Phys. 1989 527 543

[9] T. C. Brown Canad. Math. Bull. 1991 36 41

[10] S. Cantat Duke Math. J. 2009 411 460

[11] R. Carmona, J. Lacroix. Spectral theory of random Schrödinger operators. Boston: Birkhäuser, 1990.

[12] M. Casdagli Commun. Math. Phys. 1986 295 318

[13] M.-D. Choi, G. A. Elliottt, N. Yui Invent. Math. 1990 225 246

[14] D. Crisp, W. Moran, A. Pollington, P. Shiue J. Théor. Nombres Bordeaux 1993 123 137

[15] H. L. Cycon, R. G. Froese, W. Kirsch, B. Simon. Schrödinger operators. Books and monographs in physics. Berlin, Heidelberg, New York: Springer, 1987.

[16] J. M. Dahl. The spectrum of the off-diagonal Fibonacci operator. Ph.D. thesis, Rice University, 2010-2011.

[17] D. Damanik J. Math. Anal. App. 2000 393 411

[18] D. Damanik Annal. Henri Poincaré 2001 101 108

[19] D. Damanik. Strictly ergodic subshifts and associated operators. Spectral theory and mathematical physics: a Festschrift in honor of Barry Simon’s 60th birthday, Sympos. Pure Math., 76, Part 2, Amer. Math. Soc., Providence, RI, 2007.

[20] D. Damanik, M. Embree, A. Gorodetski. Spectral properties of the Schrödinger operators arising in the study of quasicrystals. (preprint) arXiv:1210.5753.

[21] D. Damanik, M. Embree, A. Gorodetski, S. Tcheremchantsev Commun. Math. Phys. 2008 499 516

[22] D. Damanik, A. Gorodetski Nonlinearity 2009 123 143

[23] D. Damanik, A. Gorodetski Commun. Math. Phys. 2011 221 277

[24] D. Damanik, A. Gorodetski Geom. Funct. Anal. 2012 976 989

[25] D. Damanik, A. Gorodetski, B. Solomyak. Absolutely continuous convolutions of singular measures and an application to the square fibonacci hamiltonian. preprint (arXiv:1306.4284).

[26] D. Damanik, P. Munger, W. N. Yessen J. Approx. Theory 2013 56 88

[27] D. Damanik, P. Munger, W. N. Yessen J. Stat. Phys. 2013 339 362

[28] D. Damanik, D. Lenz J. Anal. Math. 2003 115 139

[29] C. R. de Oliveira. Intermediate spectral theory and quantum dynamics. Progress in Mathematical Physics, vol. 54, Birkhäuser Verlag, Basel, 2009.

[30] B. Farb, D. Margalit. A primer on mapping class groups. Princeton University Press, Princeton, NJ., 2012.

[31] A. Fathi, F. Laudenbach, V. Poénaru. Travaux de Thurston sur les surfaces. Asterisque, 66, 67 (1979), (Translation by Kim, D. and Margalit, D., Thurston’s work on surfaces, Princeton University Press, 2012).

[32] N. P. Fogg. Substitutions in dynamics, arithmetics and combinatorics. Lecture Notes in Mathematics, vol. 1794, Springer-Verlag, Berlin, 2002, Edited by V. Berthé, S. Ferenczi, C. Mauduit and A. Siegel.

[33] W. H. Gottschalk Trans. Amer. Math. Soc. 1963 467 491

[34] B. C. Hall. Quantum theory for mathematicians. Graduate Texts in Mathematics, vol. 267, Springer, New York, 2013.

[35] E. Hamza, R. Sims, G. Stolz Comm. Math. Phys. 2012 215 239

[36] P. G. Harper Proc. Phys. Soc. A 1955 874 878

[37] B. Hasselblatt. Handbook of Dynamical Systems: Hyperbolic Dynamical Systems. vol. 1A, Elsevier B. V., Amsterdam, The Netherlands, 2002.

[38] B. Hasselblatt, A. Katok. Handbook of Dynamical Systems: Principal Structures. vol. 1A, Elsevier B. V., Amsterdam, The Netherlands, 2002.

[39] B. Hasselblatt, Ya. Pesin. Partially hyperbolic dynamical systems. Handbook of dynamical systems, 1B (2006), 1–55, Elsevier B. V., Amsterdam (Reviewer: C. A. Morales).

[40] M. W. Hirsch, C. C. Pugh Proc. Symp. Pure Math. 1968 133 163

[41] A. Hof J. Stat. Phys. 1993 1353 1374

[42] A. Katok, B. Hasselblatt. Introduction to the Modern Theory of Dynamical Systems. Cambridge University Press, New York, NY, 1995.

[43] M. Kohmoto, L. P. Kadanoff, C. Tang Phys. Rev. Lett. 1983 1870 1872

[44] S. Kotani Rev. Math. Phys. 1989 129 133

[45] D. Lenz Comm. Math. Phys. 2002 119 130

[46] D. Lenz, P. Stollmann J. Anal. Math. 2005 1 24

[47] E. Lieb, T. Schultz, D. Mattis Ann. Phys. 1961 407 466

[48] Q.-H. Liu, J. Peyrière, Z.-Y. Wen C. R. Math. Acad. Sci. Paris 2007 667 672

[49] Q.-H. Liu, B. Tan, Z.-X. Wen, J. Wu J. Statist. Phys. 2002 681 691

[50] J. M. Luttinger Phys. Rev. 1951 814 817

[51] R. Mañé Boletim da Sociedade Brasileira de Matemática 1990 1 24

[52] M. Mei. Spectral properties of discrete Schrödinger operators with primitive invertible substitution potentials. preprint (arXiv:1311.0954) (2013).

[53] M. Morse, G. A. Hedlund Amer. J. Math. 1940 1 42

[54] S. Newhouse. Nondensity of Axiom A on S2. Global Analysis (Proc. Sympos. Pure Math., Vol. XIV, Berkeley, Calif., 1968), 191–202, Amer. Math. Soc., Providence, RI, 1970.

[55] S. Ostlund, R. Pandit, D. Rand, H. J. Schellnhuber, E. D. Siggia Phys. Rev. Lett. 1983 1873 1876

[56] J. C. Oxtoby Bull. Amer. Math. Soc. 1952 116 136

[57] J. Palis, F. Takens. Hyperbolicity and Sensetive Chaotic Dynamics at Homoclinic Bifurcations. Cambridge University Press, Cambridge, 1993.

[58] L. Pastur, A. Figotin. Spectra of random and almost-periodic operators. Grundlehren der mathematischen Wissenschaften, Vol. 297, Springer, 1992.

[59] R. C. Penner Trans. Amer. Math. Soc. 1988 179 197

[60] Ya. Pesin. Dimension Theory in Dynamical Systems. Chicago Lect. Math. Series, 1997.

[61] Ya. Pesin. Lectures on Partial Hyperbolicity and Stable Ergodicity. Zurich Lect. Adv. Math., European Mathematical Society, 2004.

[62] M. Pollicott. Analyticity of dimensions for hyperbolic surface diffeomorphisms. preprint.

[63] C. Pugh, M. Shub, A. Wilkinson Duke Math. J. 1997 517 546

[64] L. Raymond. A constructive gap labelling for the discrete Schrödinger operator on a quasiperiodic chain. preprint (1997).

[65] C. Remling Ann. Math. 2011 125 171

[66] J. A. G. Roberts Physica A: Stat. Mech. App. 1996 295 325

[67] D. Schechtman, I. Blech, J. W. Gratias, D. Cahn Phys. Rev. Lett. 1984 1951 1953

[68] B. Simon Inverse problems and imaging 2007 713 772

[69] S. Smale Bull. Amer. Math. Soc. 1967 747 817

[70] A. Sütő Commun. Math. Phys. 1987 409 415

[71] L. A. Takhtajan. Quantum mechanics for mathematicians, Graduate Studies in Mathematics, vol. 95, American Mathematical Society, Providence, RI, 2008.

[72] B. Tan, Z.-X. Wen, Y. Zhang C. R. Math. Acad. Sci. Paris 2003 111 116

[73] G. Teschl. Jacobi operators and completely integrable nonlinear lattices. AMS mathematical surveys and monographs, vol. 72, American Mathematical Society, Providence, RI.

[74] G. Teschl. Mathematical methods in quantum mechanics. Graduate Studies in Mathematics, vol. 99, American Mathematical Society, Providence, RI, 2009, With applications to Schrödinger operators.

[75] W. P. Thurston Bull. Amer. Math. Soc. 1988 417 431

[76] M. Toda. Theory of Nonlinear Lattices. Solid-State Sciences 20, Berlin-Heidelberg-New York, Springer-Verlag, 1981.

[77] Z. Wen, Y. Zhang. Some remarks on invertible substitutions on three letter alphabet. Chinese Sci. Bull., 44 (1999).

[78] W. N. Yessen J. Spectr. Theory 2013 101 128

[79] W. N. Yessen Annal. H. Poincaré 2014 419 467

[80] W. N. Yessen Ann. Henri Poincaré 2014 793 828