Timeline for Decay of Eigenfunctions for the 1D Discrete Random Schrodinger Operators
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jul 7, 2014 at 21:38 | comment | added | Christian Remling | Yes, Oseledec sounds like the kind of ingredient that's needed here. | |
| Jul 7, 2014 at 19:08 | comment | added | Ben | Great. I'll take a look at that reference. I'm also just wondering if it's a quick step from knowing that for a fixed eigenvalue $E$, a positive Lyapunov exponent implies that the corresponding eigenfunction is exponentially decaying. Is this just the Ruelle-Osceledec theorem? | |
| Jul 7, 2014 at 18:57 | comment | added | Ben | Right. This would just give that there is no absolutely continuous part of the spectral measure. | |
| Jul 7, 2014 at 18:57 | comment | added | Christian Remling | A reference you might useful (or not, it's not exactly what you asked) is math.caltech.edu/SimonPapers/250.pdf (see Section 7 especially). | |
| Jul 7, 2014 at 18:56 | comment | added | Christian Remling | No, positive Lyapunov exponent will not give this. By Fubini, you obtain that almost surely, for almost all $E$ you have exponential decay, but the (singular) spectral measure could be supported on the exceptional set. | |
| Jul 7, 2014 at 18:50 | comment | added | Ben | Thanks! Does this follow just from the fact that the Lyapunov exponent of the transfer matrix is non-zero? In other words, if I knew that with probability one, the Lyapunov exponent is non-zero Lebesgue almost everywhere, would this imply the exponential decay for all eigenvectors? | |
| Jul 7, 2014 at 18:45 | comment | added | Christian Remling | Your last paragraph is correct, in the following sense: Fix $E$. Then $|\psi(n)|\lesssim e^{-\gamma |n|}$ almost surely. Note, however, that this on its own does not imply that almost surely, we have exponential decay for eigenvalues $E$ (this needs to be proved separately). | |
| Jul 7, 2014 at 18:39 | history | asked | Ben | CC BY-SA 3.0 |