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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 |
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). Google Scholar |
[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). Google Scholar |
[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., (). Google Scholar |
[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, (). Google Scholar |
[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). Google Scholar |
[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). Google Scholar |
[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., (). Google Scholar |
[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, (). Google Scholar |
[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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