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October  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

Steffen Klassert 1, , Daniel Lenz 2, and  Peter Stollmann 1,

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.
Keywords: absolutely continuous spectrum., Delone sets, Quasicrystals.
Mathematics Subject Classification: Primary:81Q10, 35J10; Secondary: 52C32, 82D3.
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, 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), 395-417. doi: 10.1007/s00020-004-1352-2.  Google Scholar

[2]

J. Bellissard, B. Iochum, E. Scoppola and D. Testard, Spectral properties of one-dimensional quasi-crystals, Commun. Math. Phys., 125 (1989), 527-543. doi: 10.1007/BF01218415.  Google Scholar

[3]

M. Baake and D. Lenz, Deformation of Delone dynamical systems and pure point diffraction, J. Fourier Anal. Appl., 11 (2005), 125-150. doi: 10.1007/s00041-005-4021-1.  Google Scholar

[4]

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

[5]

J. Breuer and R. Frank, Singular spectrum for radial trees, preprint, Rev. Math. Phys., 21 (2009), 929-945. doi: 10.1142/S0129055X09003773.  Google Scholar

[6]

R. Carmona and J. Lacroix, "Spectral Theory of Random Schrödinger Operators," Probability and Its Applications, Birkhäuser Boston, Inc., Boston, MA, 1990.  Google Scholar

[7]

D. Damanik, Gordon-type arguments in the spectral theory of one-dimensional quasicrystals, in "Directions in Mathematical Quasicrystals," CRM Monogr. Ser., 13, Amer. Math. Soc., Providence, RI, (2000), 277-305.  Google Scholar

[8]

D. Damanik and D. Lenz, Half-line eigenfunction estimates and purely singular continuous spectrum of zero Lebesgue measure, Forum Math., 16 (2004), 109-128. doi: 10.1515/form.2004.001.  Google Scholar

[9]

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

[10]

D. Damanik, R. Sims and G. Stolz, Localization for one-dimensional, continuum, Bernoulli-Anderson models, Duke Math. J., 114 (2002), 59-100. doi: 10.1215/S0012-7094-02-11414-8.  Google Scholar

[11]

W. G. Faris, "Self-adjoint Operators," Lecture Notes in Mathematics, Vol. 433, Springer-Verlag, Berlin-New York, 1975.  Google Scholar

[12]

A. Gordon, The point spectrum of the one-dimensional Schrödinger operator, Uspehi Mat. Nauk, 31 (1976), 257-258.  Google Scholar

[13]

C. Janot, "Quasicrystals: A Primer," Oxford University Press, Oxford, 1992. Google Scholar

[14]

M. Kaminaga, Absence of point spectrum for a class of discrete Schrödinger operators with quasiperiodic potential, Forum Math., 8 (1996), 63-69. doi: 10.1515/form.1996.8.63.  Google Scholar

[15]

S. Klassert, "Spektraltheoretische Untersuchungen von zufälligen Operatoren auf Delone-Mengen," Dissertation 2007, http://archiv.tu-chemnitz.de/pub/2007/0068/index.html. Google Scholar

[16]

S. Kotani, Ljapunov indices determine absolutely continuous spectra of stationary random one-dimensional Schrödinger operators, in "Stochastic Analysis" (Katata/Kyoto, 1982), North-Holland Math. Library, 32, North-Holland, Amsterdam, (1984), 225-247.  Google Scholar

[17]

S. Kotani, Jacobi matrices with random potentials taking finitely many values, Rev. Math. Phys., 1 (1989), 129-133. doi: 10.1142/S0129055X89000067.  Google Scholar

[18]

P. Kuchment, Quantum graphs. I. Some basic structures, Special section on quantum graphs, Waves Random Media, 14 (2004), S107-S128.  Google Scholar

[19]

J. Lagarias, Geometric models for quasicrystals I. Delone sets of finite type, Discrete Comput. Geom., 21 (1999), 161-191. doi: 10.1007/PL00009413.  Google Scholar

[20]

Y. Last and B. Simon, Eigenfunctions, transfer matrices, and absolutely continuous spectrum of one-dimensional Schrödinger operators, Invent. Math., 135 (1999), 329-367. doi: 10.1007/s002220050288.  Google Scholar

[21]

J.-Y. Lee, R. V. Moody and B. Solomyak, Pure point dynamical and diffraction spectra, Ann. Henri Poincaré, 3 (2002), 1003-1018. doi: 10.1007/s00023-002-8646-1.  Google Scholar

[22]

D. Lenz, Ergodic theory and discrete one-dimensional random Schrödinger operators: Uniform existence of the Lyapunov exponent, Contemporary Mathematics, 327 (2003), 223-238.  Google Scholar

[23]

D. Lenz, Aperiodic order via dynamical systems: Diffraction theory for sets of finite local complexity, Contemp. Math., 485, (Idris Assani, ed), (2009), 91-112.  Google Scholar

[24]

D. Lenz and P. Stollmann, Delone dynamical systems and associated random operators, in "Proc. OAMP, Constanta 2001" (eds. Combes et al), Theta Foundation, 2003.  Google Scholar

[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), 203-217. doi: 10.1215/S0012-7094-06-13121-6.  Google Scholar

[26]

D. Lenz and P. Stollmann, Generic subsets in spaces of measures and singular continuous spectrum, in "Mathematical Physics of Quantum Mechanics," Lecture Notes in Phys., 690, Springer, Berlin, (2006), 333-341  Google Scholar

[27]

M. Lothaire, "Combinatorics on Words," Encyclopedia of Mathematics and Its Applications, 17, Addison-Wesley, Reading, Massachusetts, 1983.  Google Scholar

[28]

B. Mossé, Puissances de mots et reconnaissabilité des points fixes d'une substitution, Theoret. Comput. Sci., 99 (1992), 327-334. doi: 10.1016/0304-3975(92)90357-L.  Google Scholar

[29]

M. Reed and B. Simon, "Methods of Modern Mathematical Physics. II. Fourier Analysis, Self-Adjointness," Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975.  Google Scholar

[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), 359-373. doi: 10.1007/s11040-008-9036-9.  Google Scholar

[32]

C. Seifert, in, preparation, ().   Google Scholar

[33]

M. Senechal, "Quasicrystals and Geometry," Cambridge University Press, Cambridge, 1995.  Google Scholar

[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), 1951-1953. doi: 10.1103/PhysRevLett.53.1951.  Google Scholar

[35]

B. Solomyak, Nonperiodicity implies unique composition for self-similar translationally finite tilings, Discrete Comput. Geom., 20 (1998), 265-279. doi: 10.1007/PL00009386.  Google Scholar

[36]

P. Stollmann, Smooth perturbations of regular Dirichlet forms, Proc. Amer. Math. Soc., 116 (1992), 747-752.  Google Scholar

[37]

P. Stollmann and J. Voigt, Perturbation of Dirichlet forms by measures, Potential Anal., 5 (1996), 109-138. doi: 10.1007/BF00396775.  Google Scholar

[38]

A. Sütö, The spectrum of a quasiperiodic Schrödinger operator, Commun. Math. Phys., 111 (1987), 409-415. doi: 10.1007/BF01238906.  Google Scholar

[39]

A. Sütö, Singular continuous spectrum on a Cantor set of zero Lebesgue measure for the Fibonacci Hamiltonian, J. Stat. Phys., 56 (1989), 525-531. doi: 10.1007/BF01044450.  Google Scholar

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), 395-417. doi: 10.1007/s00020-004-1352-2.  Google Scholar

[2]

J. Bellissard, B. Iochum, E. Scoppola and D. Testard, Spectral properties of one-dimensional quasi-crystals, Commun. Math. Phys., 125 (1989), 527-543. doi: 10.1007/BF01218415.  Google Scholar

[3]

M. Baake and D. Lenz, Deformation of Delone dynamical systems and pure point diffraction, J. Fourier Anal. Appl., 11 (2005), 125-150. doi: 10.1007/s00041-005-4021-1.  Google Scholar

[4]

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

[5]

J. Breuer and R. Frank, Singular spectrum for radial trees, preprint, Rev. Math. Phys., 21 (2009), 929-945. doi: 10.1142/S0129055X09003773.  Google Scholar

[6]

R. Carmona and J. Lacroix, "Spectral Theory of Random Schrödinger Operators," Probability and Its Applications, Birkhäuser Boston, Inc., Boston, MA, 1990.  Google Scholar

[7]

D. Damanik, Gordon-type arguments in the spectral theory of one-dimensional quasicrystals, in "Directions in Mathematical Quasicrystals," CRM Monogr. Ser., 13, Amer. Math. Soc., Providence, RI, (2000), 277-305.  Google Scholar

[8]

D. Damanik and D. Lenz, Half-line eigenfunction estimates and purely singular continuous spectrum of zero Lebesgue measure, Forum Math., 16 (2004), 109-128. doi: 10.1515/form.2004.001.  Google Scholar

[9]

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

[10]

D. Damanik, R. Sims and G. Stolz, Localization for one-dimensional, continuum, Bernoulli-Anderson models, Duke Math. J., 114 (2002), 59-100. doi: 10.1215/S0012-7094-02-11414-8.  Google Scholar

[11]

W. G. Faris, "Self-adjoint Operators," Lecture Notes in Mathematics, Vol. 433, Springer-Verlag, Berlin-New York, 1975.  Google Scholar

[12]

A. Gordon, The point spectrum of the one-dimensional Schrödinger operator, Uspehi Mat. Nauk, 31 (1976), 257-258.  Google Scholar

[13]

C. Janot, "Quasicrystals: A Primer," Oxford University Press, Oxford, 1992. Google Scholar

[14]

M. Kaminaga, Absence of point spectrum for a class of discrete Schrödinger operators with quasiperiodic potential, Forum Math., 8 (1996), 63-69. doi: 10.1515/form.1996.8.63.  Google Scholar

[15]

S. Klassert, "Spektraltheoretische Untersuchungen von zufälligen Operatoren auf Delone-Mengen," Dissertation 2007, http://archiv.tu-chemnitz.de/pub/2007/0068/index.html. Google Scholar

[16]

S. Kotani, Ljapunov indices determine absolutely continuous spectra of stationary random one-dimensional Schrödinger operators, in "Stochastic Analysis" (Katata/Kyoto, 1982), North-Holland Math. Library, 32, North-Holland, Amsterdam, (1984), 225-247.  Google Scholar

[17]

S. Kotani, Jacobi matrices with random potentials taking finitely many values, Rev. Math. Phys., 1 (1989), 129-133. doi: 10.1142/S0129055X89000067.  Google Scholar

[18]

P. Kuchment, Quantum graphs. I. Some basic structures, Special section on quantum graphs, Waves Random Media, 14 (2004), S107-S128.  Google Scholar

[19]

J. Lagarias, Geometric models for quasicrystals I. Delone sets of finite type, Discrete Comput. Geom., 21 (1999), 161-191. doi: 10.1007/PL00009413.  Google Scholar

[20]

Y. Last and B. Simon, Eigenfunctions, transfer matrices, and absolutely continuous spectrum of one-dimensional Schrödinger operators, Invent. Math., 135 (1999), 329-367. doi: 10.1007/s002220050288.  Google Scholar

[21]

J.-Y. Lee, R. V. Moody and B. Solomyak, Pure point dynamical and diffraction spectra, Ann. Henri Poincaré, 3 (2002), 1003-1018. doi: 10.1007/s00023-002-8646-1.  Google Scholar

[22]

D. Lenz, Ergodic theory and discrete one-dimensional random Schrödinger operators: Uniform existence of the Lyapunov exponent, Contemporary Mathematics, 327 (2003), 223-238.  Google Scholar

[23]

D. Lenz, Aperiodic order via dynamical systems: Diffraction theory for sets of finite local complexity, Contemp. Math., 485, (Idris Assani, ed), (2009), 91-112.  Google Scholar

[24]

D. Lenz and P. Stollmann, Delone dynamical systems and associated random operators, in "Proc. OAMP, Constanta 2001" (eds. Combes et al), Theta Foundation, 2003.  Google Scholar

[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), 203-217. doi: 10.1215/S0012-7094-06-13121-6.  Google Scholar

[26]

D. Lenz and P. Stollmann, Generic subsets in spaces of measures and singular continuous spectrum, in "Mathematical Physics of Quantum Mechanics," Lecture Notes in Phys., 690, Springer, Berlin, (2006), 333-341  Google Scholar

[27]

M. Lothaire, "Combinatorics on Words," Encyclopedia of Mathematics and Its Applications, 17, Addison-Wesley, Reading, Massachusetts, 1983.  Google Scholar

[28]

B. Mossé, Puissances de mots et reconnaissabilité des points fixes d'une substitution, Theoret. Comput. Sci., 99 (1992), 327-334. doi: 10.1016/0304-3975(92)90357-L.  Google Scholar

[29]

M. Reed and B. Simon, "Methods of Modern Mathematical Physics. II. Fourier Analysis, Self-Adjointness," Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975.  Google Scholar

[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), 359-373. doi: 10.1007/s11040-008-9036-9.  Google Scholar

[32]

C. Seifert, in, preparation, ().   Google Scholar

[33]

M. Senechal, "Quasicrystals and Geometry," Cambridge University Press, Cambridge, 1995.  Google Scholar

[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), 1951-1953. doi: 10.1103/PhysRevLett.53.1951.  Google Scholar

[35]

B. Solomyak, Nonperiodicity implies unique composition for self-similar translationally finite tilings, Discrete Comput. Geom., 20 (1998), 265-279. doi: 10.1007/PL00009386.  Google Scholar

[36]

P. Stollmann, Smooth perturbations of regular Dirichlet forms, Proc. Amer. Math. Soc., 116 (1992), 747-752.  Google Scholar

[37]

P. Stollmann and J. Voigt, Perturbation of Dirichlet forms by measures, Potential Anal., 5 (1996), 109-138. doi: 10.1007/BF00396775.  Google Scholar

[38]

A. Sütö, The spectrum of a quasiperiodic Schrödinger operator, Commun. Math. Phys., 111 (1987), 409-415. doi: 10.1007/BF01238906.  Google Scholar

[39]

A. Sütö, Singular continuous spectrum on a Cantor set of zero Lebesgue measure for the Fibonacci Hamiltonian, J. Stat. Phys., 56 (1989), 525-531. doi: 10.1007/BF01044450.  Google Scholar

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