The delocalized phase of the Anderson Hamiltonian in 1-D

Abstract

We introduce a random differential operator that we call ${\mathtt{CS}}_{\mathit{\tau }}$ operator, whose spectrum is given by the ${Sch}_{\mathit{\tau }}$ point process introduced by Kritchevski, Valkó and Virág (Comm. Math Phys. (2012) 314 775–806) and whose eigenvectors match with the description provided by Rifkind and Virág (Geom. Funct. Anal. (2018) 28 1394–1419). This operator acts on ${\mathbf{R}}^2$-valued functions from the interval $[0,1]$ and takes the form

where $\mathit{d}\mathcal{B}$, $\mathit{d}{\mathcal{W}}_1$ and $\mathit{d}{\mathcal{W}}_2$ are independent white noises. Then we investigate the high part of the spectrum of the Anderson Hamiltonian ${\mathcal{H}}_{\mathit{L}}:=-{\partial }_{\mathit{t}}^2+\mathit{d}\mathit{B}$ on the segment $[0,\mathit{L}]$ with white noise potential $\mathit{d}\mathit{B}$, when $\mathit{L}\to \infty$. We show that the operator ${\mathcal{H}}_{\mathit{L}}$, recentred around energy levels $\mathit{E}\sim \mathit{L}/\mathit{\tau }$ and unitarily transformed, converges in law as $\mathit{L}\to \infty$ to ${\mathtt{CS}}_{\mathit{\tau }}$ in an appropriate sense. This allows us to answer a conjecture of Rifkind and Virág on the behavior of the eigenvectors of ${\mathcal{H}}_{\mathit{L}}$. Our approach also explains how such an operator arises in the limit of ${\mathcal{H}}_{\mathit{L}}$. Finally we show that, at higher energy levels, the Anderson Hamiltonian matches (asymptotically in L) with the unperturbed Laplacian $-{\partial }_{\mathit{t}}^2$. 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.