The delocalized phase of the Anderson Hamiltonian in 1-D. (English) Zbl 1532.60147

Dumaz, Laure; Labbé, Cyril

Ann. Probab. 51, No. 3, 805-839 (2023).

Summary: We introduce a random differential operator that we call \(\mathtt{CS}_{\tau}\) operator, whose spectrum is given by the \(\operatorname{Sch}_{\tau}\) point process introduced by E. Kritchevski et al. [Commun. Math. Phys. 314, No. 3, 775–806 (2012; Zbl 1257.82059)] and whose eigenvectors match with the description provided by B. Rifkind and B. Virág [Geom. Funct. Anal. 28, No. 5, 1394–1419 (2018; Zbl 1459.60013)]. This operator acts on \(\mathbb{R}^2\)-valued functions from the interval \([0,1]\) and takes the form \[ 2\begin{pmatrix} 0 & -\partial_t \\ \partial_t & 0 \end{pmatrix} + \sqrt{\tau} \begin{pmatrix} d\mathcal{B}+\cfrac{1}{\sqrt{2}} d{\mathcal{W}_1} & \cfrac{1}{\sqrt{2}} d{\mathcal{W}_2} \\ \cfrac{1}{\sqrt{2}} d{\mathcal{W}_2} & d\mathcal{B}-\cfrac{1}{\sqrt{2}} d{\mathcal{W}_1} \end{pmatrix}, \] 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 \(\mathtt{CS}_{\tau}\) in an appropriate sense. This allows us to answer a conjecture of Rifkind and Virág on the behavior of the eigenvectors of \(\mathcal{H}_L\). 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.

MSC:

60H25	Random operators and equations (aspects of stochastic analysis)
60J60	Diffusion processes
60B20	Random matrices (probabilistic aspects)

Keywords:

Anderson Hamiltonian; canonical systems; delocalization; diffusion; Dirac operator; Hill’s operator; strong resolvent convergence

Citations:

Zbl 1257.82059; Zbl 1459.60013

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