## 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

$$2\left(\begin{array}{cc}0& -{\partial}_{\mathit{t}}\\ {\partial}_{\mathit{t}}& 0\end{array}\right)+\sqrt{\mathit{\tau}}\left(\begin{array}{cc}\mathit{d}\mathcal{B}+\frac{1}{\sqrt{2}}\phantom{\rule{0.1667em}{0ex}}\mathit{d}{\mathcal{W}}_{1}& \frac{1}{\sqrt{2}}\phantom{\rule{0.1667em}{0ex}}\mathit{d}{\mathcal{W}}_{2}\\ \frac{1}{\sqrt{2}}\phantom{\rule{0.1667em}{0ex}}\mathit{d}{\mathcal{W}}_{2}& \mathit{d}\mathcal{B}-\frac{1}{\sqrt{2}}\phantom{\rule{0.1667em}{0ex}}\mathit{d}{\mathcal{W}}_{1}\end{array}\right),$$

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.

## Funding Statement

The work of CL is supported by the project SINGULAR ANR-16-CE40-0020-01. CL was part-time affiliated to École Normale Supérieure, DMA in 2020–2021.

## Acknowledgements

The authors thank the referees for their useful comments on the first version of this paper. Special thanks are due to one anonymous referee for pointing out the link between the operator ${\mathtt{Sch}}_{\mathit{\tau}}^{\ast}$, defined by Valkó and Virág, and the operator constructed in this paper which led to Theorem 1.4 and the new Section 2.3. The authors also thank Gaultier Lambert, Benedek Valkó and Bálint Virág for their constructive remarks and corrections on the first version of this paper.

## Citation

Laure Dumaz. Cyril Labbé. "The delocalized phase of the Anderson Hamiltonian in 1-D." Ann. Probab. 51 (3) 805 - 839, May 2023. https://doi.org/10.1214/22-AOP1591

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