Holder Continuity of the Integrated Density of States for Quasi-Periodic Schrodinger Equations and Averages of Shifts of Subharmonic Functions

2001/07/01 by Michael Goldstein, Wilhelm Schlag · 230 citations
Materials Science · Mathematics · Physics and Astronomy · #Hölder condition #Mathematical analysis #Mathematics #Nonlinear system #Physics #Quantum chaos and dynamical systems #Quantum mechanics #Quasicrystal Structures and Properties #Schrödinger's cat #Spectral Theory in Mathematical Physics #Subharmonic #Subharmonic function

paper · doi:10.2307/3062114

published in Annals of Mathematics 154(1), 155 (Princeton University)

openalex publication_date 2001/07/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/25

Abstract

Michael Goldstein, Wilhelm Schlag, Hölder Continuity of the Integrated Density of States for Quasi-Periodic Schrödinger Equations and Averages of Shifts of Subharmonic Functions, Annals of Mathematics, Second Series, Vol. 154, No. 1 (Jul., 2001), pp. 155-203

Cited by

• Localization for the one-dimensional Anderson model via positivity and large deviations for the Lyapunov exponent
• Parametric Furstenberg Theorem on random products of S L ( 2 , R ) matrices
• The Fibonacci Hamiltonian
• Uniform hyperbolicity and its relation with spectral analysis of 1D discrete Schrödinger Operators
• Homogeneity of the spectrum for quasi-periodic Schrödinger operators
• Asymptotic laws for a class of quasi-periodic Schrödinger cocycles at the lowest energy of the spectrum
• Hölder regularity of the integrated density of states for quasi-periodic long-range operators on ℓ²(\Z^d)
• The absolutely continuous spectrum of the almost Mathieu operator
• On the abominable properties of the almost Mathieu operator with well approximated frequencies
• Hölder continuity of the integrated density of states for quasi-periodic Jacobi block matrices
• Random Hamiltonians with Arbitrary Point Interactions
• Continuity of the Lyapunov exponent for quasiperiodic operators with analytic potential
• On Schroedinger operators with dynamically defined potentials
• Discontinuity of Lyapunov exponent in spaces of quasiperiodic cocycles: Smoothness vs Arithmetic
• Almost reducibility and absolute continuity I
• The dynamics of a class of quasi-periodic Schrödinger cocycles
• Large deviation estimates and Hölder regularity of the Lyapunov exponents for quasi-periodic Schrödinger cocycles
• Anderson localization for two interacting quasiperiodic particles
• Fine properties of the integrated density of states and a quantitative separation property of the Dirichlet eigenvalues
• Obstructions to the regularity of the Lyapunov exponents for non-compact random Schrödinger cocycles
• On resonances and the formation of gaps in the spectrum of quasi-periodic Schroedinger equations
• Mixed spectral types for the one frequency discrete quasi-periodic Schrödinger operator
• On the finite-size Lyapunov exponent for the Schroedinger operator with skew-shift potential
• Absolutely Continuous Spectrum of Multifrequency Quasiperiodic Schrödinger operator
• Hölder continuity of the integrated density of states for Extended Harper's Model with Liouville frequency
• Sharp \frac12-Hölder continuity of the Lyapunov exponent at the bottom of the spectrum for a class of Schrödinger cocycles
• Strong Birkhoff Ergodic Theorem for subharmonic functions with irrational shift and its application to analytic quasi-periodic cocycles
• Schrödinger Operators with Dynamically Defined Potentials: A Survey
• Regularity and convergence rates for the Lyapunov exponents of linear co-cycles
• Global theory of one-frequency Schrodinger operators I: stratified analyticity of the Lyapunov exponent and the boundary of nonuniform hyperbolicity
• Transition space for the continuity of the Lyapunov exponent of quasiperiodic Schrödinger cocycles
• Hölder continuity and Fourier asymptotics of spectral measures for 1D Schrödinger operators under exponentially decaying perturbations
• Large deviations for quasi-periodic cocycles with singularities
• Method of variations of potential of quasi-periodic Schrodinger equation
• A formula related to CMV matrices and Szego cocycles
• Continuity of Lyapunov exponents in all Hölder topologies for irreducible cocycles
• Localization for random operators with non-monotone potentials with exponentially decaying correlations
• Uniform Positivity and Continuity of Lyapunov Exponents for a Class of C² Quasiperiodic Schrödinger Cocycles 
• Localization and Cantor spectrum for quasiperiodic discrete Schrödinger operators with asymmetric, smooth, cosine-like sampling functions
• Random 2D linear cocycles II: statistical properties
• On Optimal Separation of Eigenvalues for a Quasiperiodic Jacobi Matrix
• Energy supercritical nonlinear Schrödinger equations: Quasiperiodic solutions
• Abstract Continuity Theorem for the Lyapunov Exponents of linear cocycles
• The Avalanche Principle and other estimates on Grassmann manifolds
• Uniform estimates for random matrix products and applications

Related

• Frequency dependence of Hölder continuity for quasiperiodic Schrödinger operators
• Hölder continuity of the integrated density of states for quasi-periodic Jacobi operators
• Hölder regularity of the integrated density of states for quasi-periodic long-range operators on ℓ²(\Z^d)
• Localization and Regularity of the Integrated Density of States for Schrödinger Operators on ℤ^d with C²-cosine Like Quasi-periodic Potential
• Hölder continuity of the integrated density of states for quasi-periodic Jacobi block matrices