The delocalized phase of the Anderson Hamiltonian in $1$-d

Abstract

We introduce a random differential operator, that we call the ${\mathtt{CS}}_{\tau }$ operator, whose spectrum is given by the $\mbox{Sch}_τ$ point process introduced by Kritchevski, Valkó and Virág (2012) and whose eigenvectors match with the description provided by Rifkind and Virág (2018). This operator acts on ${\mathbf{R}}^2$-valued functions from the interval $[0,1]$ and takes the form: $$2\left(\begin{array}{cc}0& -{\partial }_t\\ {\partial }_t& 0\end{array}\right)+\sqrt{\tau }\left(\begin{array}{cc}d\mathcal{B}+\frac{1}{\sqrt{2}}d{\mathcal{W}}_1& \frac{1}{\sqrt{2}}d{\mathcal{W}}_2\\ \frac{1}{\sqrt{2}}d{\mathcal{W}}_2& d\mathcal{B}-\frac{1}{\sqrt{2}}d{\mathcal{W}}_1\end{array}\right),$$ where $d\mathcal{B}$, $d{\mathcal{W}}_1$ and $d{\mathcal{W}}_2$ are independent white noises. Then, we investigate the high part of the spectrum of the Anderson Hamiltonian ${\mathcal{H}}_L:=-{\partial }_t^2+dB$ on the segment $[0,L]$ with white noise potential $dB$, when $L\to \infty$. We show that the operator ${\mathcal{H}}_L$, recentred around energy levels $E\sim L/\tau$ and unitarily transformed, converges in law as $L\to \infty$ to ${\mathtt{CS}}_{\tau }$ in an appropriate sense. This allows to answer a conjecture of Rifkind and Virág (2018) on the behavior of the eigenvectors of ${\mathcal{H}}_L$. Our approach also explains how such an operator arises in the limit of ${\mathcal{H}}_L$. Finally we show that at higher energy levels, the Anderson Hamiltonian matches (asymptotically in $L$) with the unperturbed Laplacian $-{\partial }_t^2$. In a companion paper, it is shown that at energy levels much smaller than $L$, the spectrum is localized with Poisson statistics: the present paper therefore identifies the delocalized phase of the Anderson Hamiltonian.

31 pages, 1 figure