scholarly journals Positive Lyapunov exponents for Schrödinger operators with quasi-periodic potentials

1991 ◽  
Vol 142 (3) ◽  
pp. 543-566 ◽  
Author(s):  
Eugene Sorets ◽  
Thomas Spencer
Keyword(s):  
Lyapunov Exponents ◽  
Schrödinger Operators ◽  
Schrodinger Operators ◽  
Periodic Potentials

THE ROTATION NUMBER APPROACH TO EIGENVALUES OF THE ONE-DIMENSIONAL p-LAPLACIAN WITH PERIODIC POTENTIALS

2001 ◽  
Vol 64 (1) ◽  
pp. 125-143 ◽  
Author(s):  
MEIRONG ZHANG
Keyword(s):  
Dirichlet Problem ◽  
Rotation Number ◽  
Periodic Potential ◽  
Schrödinger Operators ◽  
Negative Integer ◽  
Schrodinger Operators ◽  
One Dimensional ◽  
Higher Dimensional ◽  
Periodic Potentials ◽  
The One

The paper studies the periodic and anti-periodic eigenvalues of the one-dimensional p-Laplacian with a periodic potential. After a rotation number function ρ(λ) has been introduced, it is proved that for any non-negative integer n, the endpoints of the interval ρ−1(n/2) in ℝ yield the corresponding periodic or anti-periodic eigenvalues. However, as in the Dirichlet problem of the higher dimensional p-Laplacian, it remains open if these eigenvalues represent all periodic and anti-periodic eigenvalues. The result obtained is a partial generalization of the spectrum theory of the one-dimensional Schrödinger operators with periodic potentials.

scholarly journals Schrödinger operators with dynamically defined potentials

2016 ◽  
Vol 37 (6) ◽  
pp. 1681-1764 ◽  
Author(s):  
DAVID DAMANIK
Keyword(s):  
General Theory ◽  
Schrödinger Operators ◽  
Schrodinger Operators ◽  
Almost Periodic ◽  
One Dimensional ◽  
Quantum Dynamical ◽  
Periodic Potentials ◽  
Introductory Part ◽  
Dynamical Properties

In this survey we discuss spectral and quantum dynamical properties of discrete one-dimensional Schrödinger operators whose potentials are obtained by real-valued sampling along the orbits of an ergodic invertible transformation. After an introductory part explaining basic spectral concepts and fundamental results, we present the general theory of such operators, and then provide an overview of known results for specific classes of potentials. Here we focus primarily on the cases of random and almost periodic potentials.

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