zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Reducibility or nonuniform hyperbolicity for quasiperiodic Schrödinger cocycles. (English) Zbl 1138.47033
The properties of quasiperiodic Schrödinger cocycles are studied and the conditions of reducibility and nonuniform hyperbolicity are established. A one-dimensional quasiperiodic cocycle is considered as a pair $(\alpha ,A)\in \bbfR\times C^{r}(\bbfR/\bbfZ,SL(2,\bbfR))$ viewed as a linear skew-product: $(\alpha ,A) : \bbfR/\bbfZ\times \bbfR^{2}\to \bbfR/\bbfZ\times \bbfR^{2}$. The Schrödinger cocycle $S_{v,E}\in C^{r}(\bbfR/\bbfZ,SL(2,\bbfR))$ is then defined, where $v$ is called the potential and $E$ the energy. The corresponding quasiperiodic Schrödinger cocycle $H_{v,\alpha ,x}$ is defined and it is established that its properties are closely connected to those of the family of cocycles $(\alpha ,S_{v,E})$, $E\in \bbfR$. It is shown that for almost every frequency $\alpha \in \bbfR/\bbfQ$, for every $C^{\omega}$ potential $v:\bbfR/\bbfZ\rightarrow\bbfR$, and for almost every energy $E$, the corresponding quasiperiodic Schrödinger cocycle is either reducible or nonuniformly hyperbolic. Several interesting results are emphasized: (i) zero Lebesgue measure of the singular continuous part of the spectrum of $H$ for almost every frequency, (ii) persistence of absolutely continuous spectrum under perturbations of the potential, (iii) continuity of the Lebesgue measure of the spectrum of $H$, under perturbations of the potential. It is concluded that the results of the paper give very good control on the absolutely continuous part of the spectrum of the quasiperiodic Schrödinger operator. It allows also to complete the proof of the Aubry--Andre conjecture on the measure of the spectrum of the almost Mathieu operator.

47B80Random operators (linear)
81Q10Selfadjoint operator theory in quantum theory, including spectral analysis
37A20Orbit equivalence, cocycles, ergodic equivalence relations
37D25Nonuniformly hyperbolic systems (Lyapunov exponents, Pesin theory, etc.)
39A10Additive difference equations
47B36Jacobi (tridiagonal) operators (matrices) and generalizations
47N50Applications of operator theory in quantum physics
Full Text: DOI Euclid