English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Tools annual

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

Consider the random Schr\"odinger operator $H_n$ defined on$\{0,1,\cdots,n\}\subset\mathbb{Z}$ $$  (H_n\psi)_\ell=\psi_{\ell-1,n}+\psi_{\ell+1,n}+\sigma\frac{\omega_\ell}{a_{\ell,n}}\psi_{\ell,n},\quad\psi_0=\psi_{n+1}=0, $$ where $\sigma>0$, $\omega_\ell$ are i.i.d. randomvariables and $a_{\ell,n}$ typically has order $\sqrt{n}$ for $\ell\in[\epsilonn,(1-\epsilon)n]$ and any $\epsilon>0$. Two important cases: the vanishing case$a_{\ell,n}=\sqrt{n}$ and the decaying case $a_{\ell,n}=\sqrt{\ell}$ werestudied before in \cite{kritchevski2011scaling}. In this paper we consider moregeneral decaying profiles that lie in between these two extreme cases. Wecharacterize the scaling limit of transfer matrices and determine the pointprocess limit of eigenvalues near a fixed energy in the bulk, in terms ofsolutions to coupled SDEs. We obtain new point processes that share similarproperties to the $\text{Sech}_\tau$ process. We determine the shape profile ofeigenfunctions after a suitable re-scaling, that correspond to a uniformlychosen eigenvalue of $H_n$. We also give more description of the new pointprocesses we just defined, including the probability of small and large gapsand a variance estimate.

More scaling limits for 1d random Schr\"odinger operators with critically decaying and vanishing potentials