arXiv subjects
Publications and source records attributed to Helge Krüger.
We prove that the exponential decay rate in expectation is well defined and is equal to the Lyapunov exponent, for supercritical almost Mathieu operators with Diophantine frequencies.
I give an example of a skew-shift Schrödinger operator with positive Lyapunov exponent in the spectrum for all positive coupling constant with constant frequency. This is the first example of this kind. The proof is based on CMV operators given by the skew-shift. Further results on these are derived.
We review recent results on localization for discrete alloy-type models based on the multiscale analysis and the fractional moment method, respectively. The discrete alloy-type model is a family of Schrödinger operators $H_ω= - Δ+ V_ω$ on $\ell^2 (\ZZ^d)$ where $Δ$ is the discrete Laplacian and $V_ω$ the multiplication by the function $V_ω(x) = \sum_{k \in \ZZ^d} ω_k u(x-k)$. Here $ω_k$, $k \in \ZZ^d$, are i.i.d. random variables and $u \in \ell^1 (\ZZ^d ; \RR)$ is a so-called single-site potential. Since $u$ may change sign, certain properties of $H_ω$ depend in a non-monotone way on the random parameters $ω_k$. This requires new methods at certain stages of the localization proof.
I prove that the spectrum of a skew-shift Schrödinger operator contains larges interval in the semi-classical regime. In the semi-classical limit, these intervals approach the range of the potential.