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2011, 29(4): 1553-1571. doi: 10.3934/dcds.2011.29.1553

Delone measures of finite local complexity and applications to spectral theory of one-dimensional continuum models of quasicrystals

1. 

Fakultät für Mathematik, TU Chemnitz, D - 09107 Chemnitz, Germany, Germany

2. 

Mathematisches Institut, Friedrich-Schiller Universität, Ernst-Abbe-Platz 2, D - 07743 Jena, Germany

Received  January 2010 Revised  September 2010 Published  December 2010

We study measures on the real line and present various versions of what it means for such a measure to take only finitely many values. We then study perturbations of the Laplacian by such measures. Using Kotani-Remling theory, we show that the resulting operators have empty absolutely continuous spectrum if the measures are not periodic. When combined with Gordon type arguments this allows us to prove purely singular continuous spectrum for some continuum models of quasicrystals.
Citation: Steffen Klassert, Daniel Lenz, Peter Stollmann. Delone measures of finite local complexity and applications to spectral theory of one-dimensional continuum models of quasicrystals. Discrete & Continuous Dynamical Systems - A, 2011, 29 (4) : 1553-1571. doi: 10.3934/dcds.2011.29.1553
References:
[1]

A. Ben Amor and C. Remling, Direct and inverse spectral theory of one-dimensional Schrödinger operators with measures,, Integr. Equ. Oper. Theory, 52 (2005), 395. doi: 10.1007/s00020-004-1352-2.

[2]

J. Bellissard, B. Iochum, E. Scoppola and D. Testard, Spectral properties of one-dimensional quasi-crystals,, Commun. Math. Phys., 125 (1989), 527. doi: 10.1007/BF01218415.

[3]

M. Baake and D. Lenz, Deformation of Delone dynamical systems and pure point diffraction,, J. Fourier Anal. Appl., 11 (2005), 125. doi: 10.1007/s00041-005-4021-1.

[4]

M. Baake and R. V. Moody (eds.), "Directions in Mathematical Quasicrystals,", Amer. Math. Soc., (2000).

[5]

J. Breuer and R. Frank, Singular spectrum for radial trees, preprint,, Rev. Math. Phys., 21 (2009), 929. doi: 10.1142/S0129055X09003773.

[6]

R. Carmona and J. Lacroix, "Spectral Theory of Random Schrödinger Operators,", Probability and Its Applications, (1990).

[7]

D. Damanik, Gordon-type arguments in the spectral theory of one-dimensional quasicrystals,, in, 13 (2000), 277.

[8]

D. Damanik and D. Lenz, Half-line eigenfunction estimates and purely singular continuous spectrum of zero Lebesgue measure,, Forum Math., 16 (2004), 109. doi: 10.1515/form.2004.001.

[9]

D. Damanik and G. Stolz, A generalization of Gordon's theorem and applications to quasiperiodic Schrödinger operators,, Electron. J. Differential Equations, 2000 ().

[10]

D. Damanik, R. Sims and G. Stolz, Localization for one-dimensional, continuum, Bernoulli-Anderson models,, Duke Math. J., 114 (2002), 59. doi: 10.1215/S0012-7094-02-11414-8.

[11]

W. G. Faris, "Self-adjoint Operators,", Lecture Notes in Mathematics, 433 (1975).

[12]

A. Gordon, The point spectrum of the one-dimensional Schrödinger operator,, Uspehi Mat. Nauk, 31 (1976), 257.

[13]

C. Janot, "Quasicrystals: A Primer,", Oxford University Press, (1992).

[14]

M. Kaminaga, Absence of point spectrum for a class of discrete Schrödinger operators with quasiperiodic potential,, Forum Math., 8 (1996), 63. doi: 10.1515/form.1996.8.63.

[15]

S. Klassert, "Spektraltheoretische Untersuchungen von zufälligen Operatoren auf Delone-Mengen,", Dissertation 2007, (2007).

[16]

S. Kotani, Ljapunov indices determine absolutely continuous spectra of stationary random one-dimensional Schrödinger operators,, in, 32 (1984), 225.

[17]

S. Kotani, Jacobi matrices with random potentials taking finitely many values,, Rev. Math. Phys., 1 (1989), 129. doi: 10.1142/S0129055X89000067.

[18]

P. Kuchment, Quantum graphs. I. Some basic structures,, Special section on quantum graphs, 14 (2004).

[19]

J. Lagarias, Geometric models for quasicrystals I. Delone sets of finite type,, Discrete Comput. Geom., 21 (1999), 161. doi: 10.1007/PL00009413.

[20]

Y. Last and B. Simon, Eigenfunctions, transfer matrices, and absolutely continuous spectrum of one-dimensional Schrödinger operators,, Invent. Math., 135 (1999), 329. doi: 10.1007/s002220050288.

[21]

J.-Y. Lee, R. V. Moody and B. Solomyak, Pure point dynamical and diffraction spectra,, Ann. Henri Poincaré, 3 (2002), 1003. doi: 10.1007/s00023-002-8646-1.

[22]

D. Lenz, Ergodic theory and discrete one-dimensional random Schrödinger operators: Uniform existence of the Lyapunov exponent,, Contemporary Mathematics, 327 (2003), 223.

[23]

D. Lenz, Aperiodic order via dynamical systems: Diffraction theory for sets of finite local complexity,, Contemp. Math., 485 (2009), 91.

[24]

D. Lenz and P. Stollmann, Delone dynamical systems and associated random operators,, in, (2001).

[25]

D. Lenz and P. Stollmann, Generic sets in spaces of measures and generic singular continuous spectrum for Delone Hamiltonians,, Duke Math. J., 131 (2006), 203. doi: 10.1215/S0012-7094-06-13121-6.

[26]

D. Lenz and P. Stollmann, Generic subsets in spaces of measures and singular continuous spectrum,, in, 690 (2006), 333.

[27]

M. Lothaire, "Combinatorics on Words,", Encyclopedia of Mathematics and Its Applications, 17 (1983).

[28]

B. Mossé, Puissances de mots et reconnaissabilité des points fixes d'une substitution,, Theoret. Comput. Sci., 99 (1992), 327. doi: 10.1016/0304-3975(92)90357-L.

[29]

M. Reed and B. Simon, "Methods of Modern Mathematical Physics. II. Fourier Analysis, Self-Adjointness,", Academic Press [Harcourt Brace Jovanovich, (1975).

[30]

C. Remling, The absolutely continuous spectrum of Jacobi matrices,, Annals of Math., ().

[31]

C. Remling, The absolutely continuous spectrum of one-dimensional Schrödinger operators,, Math. Phys. Anal. Geom., 10 (2007), 359. doi: 10.1007/s11040-008-9036-9.

[32]

C. Seifert, in, preparation, ().

[33]

M. Senechal, "Quasicrystals and Geometry,", Cambridge University Press, (1995).

[34]

D. Shechtman, I. Blech, D. Gratias and J. W. Cahn, Metallic phase with long-range orientational order and no translational symmetry,, Phys. Rev. Lett., 53 (1984), 1951. doi: 10.1103/PhysRevLett.53.1951.

[35]

B. Solomyak, Nonperiodicity implies unique composition for self-similar translationally finite tilings,, Discrete Comput. Geom., 20 (1998), 265. doi: 10.1007/PL00009386.

[36]

P. Stollmann, Smooth perturbations of regular Dirichlet forms,, Proc. Amer. Math. Soc., 116 (1992), 747.

[37]

P. Stollmann and J. Voigt, Perturbation of Dirichlet forms by measures,, Potential Anal., 5 (1996), 109. doi: 10.1007/BF00396775.

[38]

A. Sütö, The spectrum of a quasiperiodic Schrödinger operator,, Commun. Math. Phys., 111 (1987), 409. doi: 10.1007/BF01238906.

[39]

A. Sütö, Singular continuous spectrum on a Cantor set of zero Lebesgue measure for the Fibonacci Hamiltonian,, J. Stat. Phys., 56 (1989), 525. doi: 10.1007/BF01044450.

show all references

References:
[1]

A. Ben Amor and C. Remling, Direct and inverse spectral theory of one-dimensional Schrödinger operators with measures,, Integr. Equ. Oper. Theory, 52 (2005), 395. doi: 10.1007/s00020-004-1352-2.

[2]

J. Bellissard, B. Iochum, E. Scoppola and D. Testard, Spectral properties of one-dimensional quasi-crystals,, Commun. Math. Phys., 125 (1989), 527. doi: 10.1007/BF01218415.

[3]

M. Baake and D. Lenz, Deformation of Delone dynamical systems and pure point diffraction,, J. Fourier Anal. Appl., 11 (2005), 125. doi: 10.1007/s00041-005-4021-1.

[4]

M. Baake and R. V. Moody (eds.), "Directions in Mathematical Quasicrystals,", Amer. Math. Soc., (2000).

[5]

J. Breuer and R. Frank, Singular spectrum for radial trees, preprint,, Rev. Math. Phys., 21 (2009), 929. doi: 10.1142/S0129055X09003773.

[6]

R. Carmona and J. Lacroix, "Spectral Theory of Random Schrödinger Operators,", Probability and Its Applications, (1990).

[7]

D. Damanik, Gordon-type arguments in the spectral theory of one-dimensional quasicrystals,, in, 13 (2000), 277.

[8]

D. Damanik and D. Lenz, Half-line eigenfunction estimates and purely singular continuous spectrum of zero Lebesgue measure,, Forum Math., 16 (2004), 109. doi: 10.1515/form.2004.001.

[9]

D. Damanik and G. Stolz, A generalization of Gordon's theorem and applications to quasiperiodic Schrödinger operators,, Electron. J. Differential Equations, 2000 ().

[10]

D. Damanik, R. Sims and G. Stolz, Localization for one-dimensional, continuum, Bernoulli-Anderson models,, Duke Math. J., 114 (2002), 59. doi: 10.1215/S0012-7094-02-11414-8.

[11]

W. G. Faris, "Self-adjoint Operators,", Lecture Notes in Mathematics, 433 (1975).

[12]

A. Gordon, The point spectrum of the one-dimensional Schrödinger operator,, Uspehi Mat. Nauk, 31 (1976), 257.

[13]

C. Janot, "Quasicrystals: A Primer,", Oxford University Press, (1992).

[14]

M. Kaminaga, Absence of point spectrum for a class of discrete Schrödinger operators with quasiperiodic potential,, Forum Math., 8 (1996), 63. doi: 10.1515/form.1996.8.63.

[15]

S. Klassert, "Spektraltheoretische Untersuchungen von zufälligen Operatoren auf Delone-Mengen,", Dissertation 2007, (2007).

[16]

S. Kotani, Ljapunov indices determine absolutely continuous spectra of stationary random one-dimensional Schrödinger operators,, in, 32 (1984), 225.

[17]

S. Kotani, Jacobi matrices with random potentials taking finitely many values,, Rev. Math. Phys., 1 (1989), 129. doi: 10.1142/S0129055X89000067.

[18]

P. Kuchment, Quantum graphs. I. Some basic structures,, Special section on quantum graphs, 14 (2004).

[19]

J. Lagarias, Geometric models for quasicrystals I. Delone sets of finite type,, Discrete Comput. Geom., 21 (1999), 161. doi: 10.1007/PL00009413.

[20]

Y. Last and B. Simon, Eigenfunctions, transfer matrices, and absolutely continuous spectrum of one-dimensional Schrödinger operators,, Invent. Math., 135 (1999), 329. doi: 10.1007/s002220050288.

[21]

J.-Y. Lee, R. V. Moody and B. Solomyak, Pure point dynamical and diffraction spectra,, Ann. Henri Poincaré, 3 (2002), 1003. doi: 10.1007/s00023-002-8646-1.

[22]

D. Lenz, Ergodic theory and discrete one-dimensional random Schrödinger operators: Uniform existence of the Lyapunov exponent,, Contemporary Mathematics, 327 (2003), 223.

[23]

D. Lenz, Aperiodic order via dynamical systems: Diffraction theory for sets of finite local complexity,, Contemp. Math., 485 (2009), 91.

[24]

D. Lenz and P. Stollmann, Delone dynamical systems and associated random operators,, in, (2001).

[25]

D. Lenz and P. Stollmann, Generic sets in spaces of measures and generic singular continuous spectrum for Delone Hamiltonians,, Duke Math. J., 131 (2006), 203. doi: 10.1215/S0012-7094-06-13121-6.

[26]

D. Lenz and P. Stollmann, Generic subsets in spaces of measures and singular continuous spectrum,, in, 690 (2006), 333.

[27]

M. Lothaire, "Combinatorics on Words,", Encyclopedia of Mathematics and Its Applications, 17 (1983).

[28]

B. Mossé, Puissances de mots et reconnaissabilité des points fixes d'une substitution,, Theoret. Comput. Sci., 99 (1992), 327. doi: 10.1016/0304-3975(92)90357-L.

[29]

M. Reed and B. Simon, "Methods of Modern Mathematical Physics. II. Fourier Analysis, Self-Adjointness,", Academic Press [Harcourt Brace Jovanovich, (1975).

[30]

C. Remling, The absolutely continuous spectrum of Jacobi matrices,, Annals of Math., ().

[31]

C. Remling, The absolutely continuous spectrum of one-dimensional Schrödinger operators,, Math. Phys. Anal. Geom., 10 (2007), 359. doi: 10.1007/s11040-008-9036-9.

[32]

C. Seifert, in, preparation, ().

[33]

M. Senechal, "Quasicrystals and Geometry,", Cambridge University Press, (1995).

[34]

D. Shechtman, I. Blech, D. Gratias and J. W. Cahn, Metallic phase with long-range orientational order and no translational symmetry,, Phys. Rev. Lett., 53 (1984), 1951. doi: 10.1103/PhysRevLett.53.1951.

[35]

B. Solomyak, Nonperiodicity implies unique composition for self-similar translationally finite tilings,, Discrete Comput. Geom., 20 (1998), 265. doi: 10.1007/PL00009386.

[36]

P. Stollmann, Smooth perturbations of regular Dirichlet forms,, Proc. Amer. Math. Soc., 116 (1992), 747.

[37]

P. Stollmann and J. Voigt, Perturbation of Dirichlet forms by measures,, Potential Anal., 5 (1996), 109. doi: 10.1007/BF00396775.

[38]

A. Sütö, The spectrum of a quasiperiodic Schrödinger operator,, Commun. Math. Phys., 111 (1987), 409. doi: 10.1007/BF01238906.

[39]

A. Sütö, Singular continuous spectrum on a Cantor set of zero Lebesgue measure for the Fibonacci Hamiltonian,, J. Stat. Phys., 56 (1989), 525. doi: 10.1007/BF01044450.

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