Large gaps asymptotics for the 1-dimensional random Schrödinger operator

Abstract

We show that in the Schrödinger point process, Sch$_\tau$, $\tau>0,$ the probability of having no eigenvalue in a fixed interval of size $\lambda$ is given by \[ \exp\left(-\frac{\lambda^2}{4\tau}+\left(\frac{2}{\tau}-\frac{1}{4}\right)\lambda +o(\lambda)\right), \]as $\lambda\to\infty.$ It is a slightly more precise version than the one given in a previous work.