The Sch$_\tau$ operator and the delocalized phase of the Anderson Hamiltonian in $1$-d

We introduce a random differential operator, that we call the $\mathtt{Sch}_\tau$ operator, whose spectrum is given by the $\mbox{Sch}_\tau$ point process introduced by Kritchevski, Valko and Virag (2012) and whose eigenvectors match with the description provided by Rifkind and Virag (2018). This operator acts on $\mathbf{R}^2$-valued functions from the interval $[0,1]$ and takes the form: $$ 2 \begin{pmatrix} 0 & -\partial_t \\ \partial_t & 0 \end{pmatrix} + \sqrt{\tau} \begin{pmatrix} d\mathcal{B} + \frac1{\sqrt 2} d\mathcal{W}_1 & \frac1{\sqrt 2} d\mathcal{W}_2\\ \frac1{\sqrt 2} d\mathcal{W}_2 & d\mathcal{B} - \frac1{\sqrt 2} d\mathcal{W}_1\end{pmatrix}\,, $$ 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 $\mathtt{Sch}_\tau$ in an appropriate sense. This allows to answer a conjecture of Rifkind and Virag (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.