Access

You are not currently logged in.

Access your personal account or get JSTOR access through your library or other institution:

login

Log in to your personal account or through your institution.

If You Use a Screen Reader

This content is available through Read Online (Free) program, which relies on page scans. Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.

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

Michael Goldstein and Wilhelm Schlag
Annals of Mathematics
Second Series, Vol. 154, No. 1 (Jul., 2001), pp. 155-203
Published by: Annals of Mathematics
DOI: 10.2307/3062114
Stable URL: http://www.jstor.org/stable/3062114
Page Count: 49
  • Read Online (Free)
  • Subscribe ($19.50)
  • Cite this Item
Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
Hölder Continuity of the Integrated Density of States for Quasi-Periodic Schrödinger Equations and Averages of Shifts of Subharmonic Functions
Preview not available

Abstract

In this paper we consider various regularity results for discrete quasi-periodic Schrödinger equations -ψn+1-ψn-1+V (θ + nω)ψn = Eψn with analytic potential V. We prove that on intervals of positivity for the Lyapunov exponent the integrated density of states is Hölder continuous in the energy provided ω has a typical continued fraction expansion. The proof is based on certain sharp large deviation theorems for the norms of the monodromy matrices and the "avalanche-principle". The latter refers to a mechanism that allows us to write the norm of a monodromy matrix as the product of the norms of many short blocks. In the multi-frequency case the integrated density of states is shown to have a modulus of continuity of the form exp(-|log t|σ) for some 0 < σ < 1, but currently we do not obtain Hölder continuity in the case of more than one frequency. We also present a mechanism for proving the positivity of the Lyapunov exponent for large disorders for a general class of equations. The only requirement for this approach is some weak form of a large deviation theorem for the Lyapunov exponents. In particular, we obtain an independent proof of the Herman-Sorets-Spencer theorem in the multi-frequency case. The approach in this paper is related to the recent nonperturbative proof of Anderson localization in the quasi-periodic case by J. Bourgain and M. Goldstein.

Page Thumbnails

  • Thumbnail: Page 
[155]
    [155]
  • Thumbnail: Page 
156
    156
  • Thumbnail: Page 
157
    157
  • Thumbnail: Page 
158
    158
  • Thumbnail: Page 
159
    159
  • Thumbnail: Page 
160
    160
  • Thumbnail: Page 
161
    161
  • Thumbnail: Page 
162
    162
  • Thumbnail: Page 
163
    163
  • Thumbnail: Page 
164
    164
  • Thumbnail: Page 
165
    165
  • Thumbnail: Page 
166
    166
  • Thumbnail: Page 
167
    167
  • Thumbnail: Page 
168
    168
  • Thumbnail: Page 
169
    169
  • Thumbnail: Page 
170
    170
  • Thumbnail: Page 
171
    171
  • Thumbnail: Page 
172
    172
  • Thumbnail: Page 
173
    173
  • Thumbnail: Page 
174
    174
  • Thumbnail: Page 
175
    175
  • Thumbnail: Page 
176
    176
  • Thumbnail: Page 
177
    177
  • Thumbnail: Page 
178
    178
  • Thumbnail: Page 
179
    179
  • Thumbnail: Page 
180
    180
  • Thumbnail: Page 
181
    181
  • Thumbnail: Page 
182
    182
  • Thumbnail: Page 
183
    183
  • Thumbnail: Page 
184
    184
  • Thumbnail: Page 
185
    185
  • Thumbnail: Page 
186
    186
  • Thumbnail: Page 
187
    187
  • Thumbnail: Page 
188
    188
  • Thumbnail: Page 
189
    189
  • Thumbnail: Page 
190
    190
  • Thumbnail: Page 
191
    191
  • Thumbnail: Page 
192
    192
  • Thumbnail: Page 
193
    193
  • Thumbnail: Page 
194
    194
  • Thumbnail: Page 
195
    195
  • Thumbnail: Page 
196
    196
  • Thumbnail: Page 
197
    197
  • Thumbnail: Page 
198
    198
  • Thumbnail: Page 
199
    199
  • Thumbnail: Page 
200
    200
  • Thumbnail: Page 
201
    201
  • Thumbnail: Page 
202
    202
  • Thumbnail: Page 
203
    203