Singular continuous spectrum for a class of nonprimitive substitution Schrödinger operators

de Oliveira, César; Lima, Marcus

doi:10.1090/S0002-9939-01-06148-2

Singular continuous spectrum for a class of nonprimitive substitution Schrödinger operators
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by César R. de Oliveira and Marcus V. Lima PDF

Proc. Amer. Math. Soc. 130 (2002), 145-156 Request permission

Abstract:

We present a class of discrete Schrödinger operators, with potentials derived from nonprimitive substitutions, that has purely singular continuous spectrum. We give sufficient conditions on the substitution rule assuring singular continuous spectrum, either for a generic set in the hull of the potential or for a set of total invariant measure.

References

Françoise Axel and Denis Gratias (eds.), Beyond quasicrystals, Springer-Verlag, Berlin; Les Éditions de Physique, Les Ulis, 1995. Papers from the Winter School held in Les Houches, March 7–18, 1994. MR 1420414, DOI 10.1007/978-3-662-03130-8
J. Bellissard, Spectral properties of Schrödinger’s operator with a Thue-Morse potential, Number theory and physics (Les Houches, 1989) Springer Proc. Phys., vol. 47, Springer, Berlin, 1990, pp. 140–150. MR 1058456, DOI 10.1007/978-3-642-75405-0_{1}3
Jean Bellissard, Anton Bovier, and Jean-Michel Ghez, Spectral properties of a tight binding Hamiltonian with period doubling potential, Comm. Math. Phys. 135 (1991), no. 2, 379–399. MR 1087389
Jean Bellissard, Anton Bovier, and Jean-Michel Ghez, Gap labelling theorems for one-dimensional discrete Schrödinger operators, Rev. Math. Phys. 4 (1992), no. 1, 1–37. MR 1160136, DOI 10.1142/S0129055X92000029
Anton Bovier and Jean-Michel Ghez, Spectral properties of one-dimensional Schrödinger operators with potentials generated by substitutions, Comm. Math. Phys. 158 (1993), no. 1, 45–66. MR 1243715
Anton Bovier and Jean-Michel Ghez, Erratum: “Spectral properties of one-dimensional Schrödinger operators with potentials generated by substitutions” [Comm. Math. Phys. 158 (1993), no. 1, 45–66; MR1243715 (94k:82064)], Comm. Math. Phys. 166 (1994), no. 2, 431–432. MR 1309558
David Damanik, Singular continuous spectrum for the period doubling Hamiltonian on a set of full measure, Comm. Math. Phys. 196 (1998), no. 2, 477–483. MR 1645023, DOI 10.1007/s002200050432
David Damanik, Singular continuous spectrum for a class of substitution Hamiltonians, Lett. Math. Phys. 46 (1998), no. 4, 303–311. MR 1668569, DOI 10.1023/A:1007510721504
François Delyon and Dimitri Petritis, Absence of localization in a class of Schrödinger operators with quasiperiodic potential, Comm. Math. Phys. 103 (1986), no. 3, 441–444. MR 832920
F. Delyon and J. Peyrière, Recurrence of the eigenstates of a Schrödinger operator with automatic potential, J. Statist. Phys. 64 (1991), no. 1-2, 363–368. MR 1117660, DOI 10.1007/BF01057881
A. Hof, Some remarks on discrete aperiodic Schrödinger operators, J. Statist. Phys. 72 (1993), no. 5-6, 1353–1374. MR 1241543, DOI 10.1007/BF01048190
A. Hof, O. Knill, and B. Simon, Singular continuous spectrum for palindromic Schrödinger operators, Comm. Math. Phys. 174 (1995), no. 1, 149–159. MR 1372804
Shinichi Kotani, Jacobi matrices with random potentials taking finitely many values, Rev. Math. Phys. 1 (1989), no. 1, 129–133. MR 1041533, DOI 10.1142/S0129055X89000067
András Sütő, The spectrum of a quasiperiodic Schrödinger operator, Comm. Math. Phys. 111 (1987), no. 3, 409–415. MR 900502
András Sütő, Singular continuous spectrum on a Cantor set of zero Lebesgue measure for the Fibonacci Hamiltonian, J. Statist. Phys. 56 (1989), no. 3-4, 525–531. MR 1009514, DOI 10.1007/BF01044450
B. Iochum and D. Testard, Power law growth for the resistance in the Fibonacci model, J. Statist. Phys. 65 (1991), no. 3-4, 715–723. MR 1137430, DOI 10.1007/BF01053750
de Oliveira, C. R., Lima, M. V.: A nonprimitive substitution Schrödinger operator with generic singular continuous spectrum. Rep. Math. Phys. 45, 431–436 (2000).
H. L. Cycon, R. G. Froese, W. Kirsch, and B. Simon, Schrödinger operators with application to quantum mechanics and global geometry, Springer Study Edition, Texts and Monographs in Physics, Springer-Verlag, Berlin, 1987. MR 883643
Martine Queffélec, Substitution dynamical systems—spectral analysis, Lecture Notes in Mathematics, vol. 1294, Springer-Verlag, Berlin, 1987. MR 924156, DOI 10.1007/BFb0081890
Zhi-Ying Wen, Singular words, invertible substitutions and local isomorphisms, Beyond quasicrystals (Les Houches, 1994) Springer, Berlin, 1995, pp. 433–440. MR 1420425, DOI 10.1007/BF02245815
Barry Simon, Operators with singular continuous spectrum. I. General operators, Ann. of Math. (2) 141 (1995), no. 1, 131–145. MR 1314033, DOI 10.2307/2118629
Yoram Last and Barry Simon, Eigenfunctions, transfer matrices, and absolutely continuous spectrum of one-dimensional Schrödinger operators, Invent. Math. 135 (1999), no. 2, 329–367. MR 1666767, DOI 10.1007/s002220050288
Karl Petersen, Ergodic theory, Cambridge Studies in Advanced Mathematics, vol. 2, Cambridge University Press, Cambridge, 1983. MR 833286, DOI 10.1017/CBO9780511608728